LIBRARY 

OF    THE 

UNIVERSITY  OF  CALIFORNIA. 
Class 


THE  MATHEMATICAL  THEORY 

;    .  .   '       .  OF          .  '/•"•• 

ECLIPSES 

ACCORDING  TO   CHAUVENETS  TRANSFORMATION   OF 

BESSEL'S  METHOD 

EXPLAINED    AND    ILLUSTRATED 

TO    WHICH    ARE    APPENDED 

TRANSITS    OF    MERCURY  AND  VENUS 
AND  OCCULTATIONS  OF  FIXED  STARS 

BY 

ROBERDEAU  BUCHANAN,  S.B. 

ASSISTANT    IN    THE     NAUTICAL     ALMANAC     OFFICE,     UNITED     STATES     NAVAL     OBSERVATORY,     AND 

COMPUTER    OF    THE    ECLIPSES    FOR    TWENTY-THREE    YEARS 

AUTHOR    OF    "A    TREATISE    ON    THE    PROJECTION    OF    THE    SPHERE,"    AND    OTHER    WORKS, 
HISTORICAL    AND    BIOGRAPHICAL 


PHILADELPHIA  VAND  LONDON 

J.  B.  LIPPINCOTT   COMPANY 

1904 


cC 


Entered  according  to  Act  of  Congress,  in  the  year  1904,  by 

EOBEEDEAU  BUCHANAN, 
In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


PREFACE. 


THE  present  work  is  designed  as  a  convenient  hand-book  for  the 
computer  of  Solar  and  Lunar  Eclipses.  It  has  originated  partly 
from  a  manuscript  book  of  formulae  and  precepts,  prepared  by  the 
author  for  his  own  use,  during  the  period  of  twenty-four  years,  in 
which,  besides  other  astronomical  work,  he  has  been  engaged  in  com- 
puting the  eclipses  for  the  American  Ephemeris  and  Nautical  Al- 
manac. The  publication  was  suggested  by  reason  of  sundry  appli- 
cations to  him,  both  orally  and  by  letter,  from  persons  desiring 
information  on  the  subject  of  eclipses,  and  also  explanations  of 
difficult  passages  in  CHAUVENET'S  Astronomy. 

The  chapter  on  Eclipses  in  this  work — Spherical  and  Practical 
Astronomy,  by  Professor  WILLIAM  CHAUVENET,  published  in  1863 
— is  the  most  thorough  and  exhaustive  treatise  on  this  subject  that 
has  yet  been  published.  It  is  taken  as  the  standard  authority  for 
the  methods  of  computation  by  both  the  English  and  the  American 
Nautical  Almanacs. 

The  theory  is  one  of  considerable  intricacy,  and  while  the  student 
can  generally  follow  CHAUVENET  in  deducing  the  formulae,  yet  he 
meets  with  some  difficulty  in  grasping  the  subject,  finding  many 
points  requiring  further  explanation  than  is  given  in  the  text.  No 
other  publication  is  of  the  least  assistance  in  explaining  CHAUVE- 
NET'S  chapter. 

There  is  another  difficulty  encountered  by  the  practical  computer, 
no  matter  how  well  skilled  he  may  be  in  mathematical  formulae  or 
the  art  of  computing ;  and  that  is,  the  formulae  in  CHAUVENET  are 
not  given  in  the  order  they  are  to  be  used,  but  just  as  they  chance 
to  be  derived.  They  are  so  numerous,  and  so  much  scattered,  that 
it  is  almost  imperatively  necessary  for  the  computer  to  write  them 
on0  in  the  order  they  are  to  be  taken  up.  This  has  been  done  in  the 
present  work.  Moreover,  some  of  the  formulae  given  by  CHAUVE- 
NET, adding  to  the  completeness  of  his  work,  are  not  absolutely 
necessary,  and  may  be  disregarded.  These  also  have  been  here 
pointed  out. 

iii 

S954 


iv  PREFACE. 

And,  further,  CHAUVENET'S  admirable  treatise  is  so  rigorously 
exact  in  all  its  formulae,  that  many  of  them,  especially  those  required 
only  for  projection  orf  a  chart,  may  be  considerably  simplified,  and 
the  labor  of  using  them  thereby  diminished.  This  has  also  been 
pointed  out  in  the  proper  places. 

Finally,  during  the  time  the  author  has  been  engaged  upon  the 
Nautical  Almanac  in  computing  the  eclipses  (which  is  longer  than 
that  of  any  of  his  predecessors  upon  this  portion  of  the  Almanac), 
the  eclipses  have  rotated  through  somewhat  more  than  one  Saros 
of  eighteen  years,  after  which  they  repeat  themselves  very  similarly. 
Their  peculiarities  have  been  here  noted,  together  with  many  details 
not  before  published  in  any  astronomy ;  difficult  passages  in  CHAUVE- 
NET'S  chapter,  so  far  as  they  have  been  called  to  the  author's  atten- 
tion, are  here  explained ;  some  errata  and  other  mistakes  in  the  text 
and  in  one  figure,  noticed ;  and  the  formulae  themselves  explained 
graphically,  with  numerous  precepts  given  for  the  practical  com- 
puter. 

The  graphic  method  here  employed  for  explaining  the  formulae  is 
a  feature  of  this  work,  which  the  author  believes  has  not  heretofore 
been  made  use  of  so  extensively  for  explaining  a  complicated  series 
of  formulae.  The  eclipse  is  dissected  after  the  manner  of  a  surgeon 
— it  is  cut  up  and  the  hidden  parts  laid  open  to  view. 

The  whole  subject  is  treated  pretty  much  as  a  lecturer  might 
address  his  class  after  closing  his  text-book. 

The  author  desires  to  express  his  obligations  especially  to  Louis 
CHAUVENET,  Esq.,  of  St.  Louis,  son  of  Professor  CHAUVENET,  and 
holder  of  the  copyright  of  his  Astronomy,  for  his  cordial  and  full 
permission  to  quote  from  his  father's  work;  and  also  to  Professor 
WALTER  S.  HARSHMAN,  U.  S.  N.,  Director  of  the  Nautical  Almanac 
office,  for  the  privilege  of  photographing  the  author's  original  draw- 
ings made  for  that  office,  of  the  solar  eclipses  of  September  9,  1904, 
and  August  8, 1896,  which  are  reproduced  in  the  present  work.  Also 
to  Professor  ASAPH  HALL,  U.  S.  N. ;  to  Professor  FRANK  H.  BIGE- 
LOW,  of  the  Weather  Bureau,  and  to  Mr.  J.  ROBERTSON,  of  the 
Nautical  Almanac  office,  for  reading  portions  of  the  manuscript  of 
this  work. 

THE  AUTHOE. 
2015  Q  STREET,  WASHINGTON,  D.  C. 
1904,  March  24. 


ANALYTICAL  TABLE  OF  CONTENTS. 


PAKT  I.— SOLAE  ECLIPSES. 

SEC.      ART.  PAGE 

Principal  Notation  with  references  for  explanation ix 

I.       1.     Introduction.     Retrospect 17 

2.  Eemarks  on  the  Computation 18 

3.  Differentiated  quantities  cannot  always  be  shown  graphically  ....    20 

4.  The  darkness  of  the  Crucifixion 21 

II.       5.     The  Criterion  for  an  eclipse 23 

8.     Rigorous  Formulae 26 

10.  Semidiameters  and  parallaxes,  corrected  values 28 

11.  TheSaros 30 

III.  15.     Data  and  Elements 33 

17.     Data  for  an  Example 34,  35 

20.  Eclipse  Constants   .    .    .    . 36 

21.  Elements 39 

IV.  22.     Eclipse  Tables 41 

25.     The  Main  Computation  and  Formulae 42  to  50 

30.    The  Epoch  Hour 45 

36.  Example  of  the  Main  Computation 51,  54 

37.  Eclipse  Tables.     Example 52-3-5 

38.  Geometrical  Illustration 54-62 

39.  Principle  of  Spherical  Projection 56 

45.  Motion  of  the  Successive  eclipses  of  one  series 60 

46.  The  Umbral  Cone  and  Species  of  the  Eclipse 62-5 

V.     50.     The  Extreme  Times  Generally.     Formula? 66 

The  Local  Apparent  time  given  by  & 67,  119,  153 

52.  Middle  of  the  Eclipse   ...       68 

53.  Criterion  for  Partial  Eclipse .    .    69 

54.  Example  of  the  Computation  with  remarks 69-70 

55.  The  internal  contacts 70 

57.  Check  formulae  in  the  Computation 72 

Another  method  of  computing  the  times 73 

58.  Geometrical  Illustration.     Extreme  times 73 

VI.    61.     Rising  and  Setting  curve 77 

61.    CHAUVENET'S  Formulae 77 

64.  Dr.  HILL'S  Formulae 79 

65.  Example .    79 

66.  The  Node  or  Multiple  point 80,  89 

67.  Singular  forms  of  the  curve 81 


vi  ANALYTICAL  TABLE  OF   CONTENTS. 

SEC.         ART.  PAGE 

VII.     68.  Maximum  Curve 82 

68.  Formulae 82 

69.  Kemarks  on  the  formulae  and  CHAUVENET'S  text 83 

70.  Maximum  Curve  and  angle  E  explained 84 

75.  Approximate  Formulae  for  general  use 87 

76.  Example 88 

77.  Relations  of  the  Node  and  Maximum  Curve 88 

78.  Greatest  Eclipse.     Partial  Eclipse 89 

79.  Magnitude 92 

VHI.      80.     Outline  of  the  Shadow 92 

80.  The  Angle  Q  explained 92 

81.  General  Formulae 94 

84.  Approximate  Formulae 96 

85.  Geometrical  Illustration 96 

87.    Dr.  HILL'S  Formulae 98 

90.  Example 102 

IX.      91.     Extreme  Times  N  and  S  Limits  of  Penumbra 104 

91.  Formulae 104 

92.  Example 105 

X.      93.     Northern  and  Southern  Limiting  Curve  (Penumbra) 107 

93.  Formula 107 

94.  Limits  for  Q,  CHAUVENET  108.    Art.  95  With  special  Tables    .    .  108 

96.  Example 110 

97.  Relations  of  the  angles  Q  and  E 110 

99.     Approximate  Formulae 112 

100.  Example 113 

101.  Curves  of  Any  degree  of  Obscuration 114 

XI.     102.    Extreme  Times  of  Central  Eclipse 115 

102.  Formulae 115 

105.     Check  on  the  Times 116 

107.  Example 117 

XII.    108.     Curve  of  Central  Eclipse 118 

108.  Formulae 118 

110.  Geometrical  Illustration 119 

111.  Example,  Central  Curve,  Duration  and  Limits 120 

XIII.  112.     Central  Eclipse  at  Noon 121 

112.  Formulae 121 

113.  Check  on  y  at  tf  and  what  it  shows 44,  121-2,  157,  195 

114.  Example 122 

115.  Special  Cases 122 

XIV.  116.     Duration  of  Total  or  Annular  Eclipse 123 

116.  Formulae 123 

117.  Example 120,124 

118.  The  Angle  Q  in  Duration 124 

XV.     119.     Extreme  Times  N.  and  S.  Limits  of  Umbra   . 125 

119.  Formulae 125 

122.     Example 127 

127.     Check  Formulae,  131.     Example 132 

129.     EiTor  in  CHAUVENET'S  Formulae 133 

134.  Corrected  Formulae  .                                                                            .  137 


ANALYTICAL  TABLE   OF   CONTENTS. 


vii 


SEC.          ART.  PAGE 

XVI.    135.    Curves,  N.  and  S.  Limits  of  Umbra 137 

135.  Kemarks 137 

136.  Formulae 138 

137.  The  angle  Q,  and  method  of  shortening  the  computation  ...    138 

138.  Example 120,  139 

139.  Peculiarities  of  this  Curve 140 

XVII.    141.     The  Chart 141 

142.     Kinds  of  projection 142 

146.    The  Drawing 143 

XVIII.    149.     Prediction  for  a  Given  Plan 144 

149.  Preliminary  reductions 144 

150.  General  Formulae 146 

151.  Example 147-9 

153.  Angles  of  Position,  Formulae 151 

154.  Partial  Eclipse,  Greatest  Eclipse 152 

155.  Duration.     Magnitude 152-3 

157.     Checks  on  the  Times 153 

XIX.    158.     Prediction  by  the  Method  of  Semidiameters 156 

159.     Projection.     Example 157-8 

166.     Partial  Eclipse 162 

168.    Computation 163 

XX.    170.     Shape  of  the  Shadow  upon  the  Earth 169 

170.  Shadow  Bands 169 

171.  The  Ellipse  of  Shadow 171 

172.  The  Axes  and  Conjugate  Diameters,  245,  Example 172 

173.  Proof  of  the  Ellipse.     Example 175 

174  /  Construction  of  the  Ellipse  Graphically \  1 77 

I  Computation,  Data,  Example • / 

175.  Computation  for  the  Tangent,  Formulae 179 

Data,  Example • 182 

176.  Check  upon  the  foregoing  Formulae,  Example 184 

177.  Width  of  the  Shadow  Path 185 

178.  Velocity  of  the  Shadow.     Example,  Width,  and  Velocity  ...    186 

180.  Rigorous  method  for  computation  of  the  Ellipse  of  Shadow   .    .    187 

181.  Circular  Shadow 187 

XXL    182.  The  Constant  k  in  Eclipses  and  Occultations,  and  their  differences  189 

182.  The  Constant  k.     Differential  Formulae 191 

183.  The  earth  considered  smaller  in  Eclipses  than  in  Occultations    .    193 
XXII.    184.     Corrections  for  Refraction  and  Altitude 196 

184.  Correction  for  Refraction 196 

185.  Correction  for  the  altitude  of  the  observer 198 

XXIII.    186.    Professor  Safford's  Transformation 198 

186.  Formulae 199 

187.  Prediction  by  this  method 200 


viii  ANALYTICAL   TABLE   OF   CONTENTS. 


PART  II. 

SEC.            ART.                                                                                                                                             •  PAGE 

XXIV.    188.    LUNAE  ECLIPSES 202 

188.  Introduction  and  Criterion      202 

189.  General  Formulae 204 

190.  Data,  Elements,  Example,  Preliminary  work 206 

191.  The  Times,  Angles  of  Position,  Magnitude 209 

192.  Lunar  Appulse 211 

193.  Graphic  Method  for  Lunar  Eclipses .  211 

XXV.    195.    TKANSITS  OF  MEKCUEY  AND  VENUS  ACEOSS  THE 

SUN'S  DISK    ...-..- 214 

195.  Data,  Elements 214 

196.  General  Formulae 215 

197.  Example,  Transit  of  Mercury,  1894 217 

198.  Graphic  method  of  representation 221 

XXVI.     200.    OCCULTATIONS  OF  FIXED  STAES  BY  THE  MOON  .   .  222 

200.  General  Formula? 222 

201.  Prediction  for  a  Given  Place 224 

202.  The  Limiting  Parallels    .    .    .    .    ; 226 

203.  Example 228 

204.  Geometrical  representation  of  Formulae 231 

205.  Limiting  Parallels  graphically  shown 233 

206.  Criterion  of  visibility  . 235 

TABLES,  with  List  and  reference  to  pages 237 

PLATES,  with  List  of  all  Plates  and  Figures,  and  Index  to  Geometrical  Con- 
structions                     .  249 


PRINCIPAL  NOTATION. 


[Other  quantities  not  generally  used  are  defined  on  the  pages  where  they  occur.] 


o 


or 


VI 

ft 

a'  6'  r 
a  6  r 
ad 


h' 
xyz 

V  2/o 
x'y' 

y\ 

II, 
L 

L  • 


?, 
c  c, 

CO, 
Cc 


The  sun. 

The  moon.     ,    : 

A  fixed  star,  Section  XXVI.,  Occultations. 

Conjunction,  at  which  time  Solar  eclipses  occur,  Arts.  12,  21. 

Opposition,  at  which  time  Lunar  eclipses  occur,  Art.  190. 

Ascending  node  of  an  orbit;  chiefly  here, that  of  the  moon,  Art.  5. 

Sign  of  Division  in  the  Examples. 

Sun  and  moon's  longitudes,  Art.  5. 

Moon's  latitude,  Art.  5. 

Sun's    |  ^  A    -p       and  Distances  from  the  earth,  Arts.  15,  17. 

Moon's ) 

Eight  ascension  and  declination  of  the  point  Z,  Arts.  25,  38. 

Transformation  of  d  to  take  account  of  the  earth's  spheroid,  Arts.  32,  42. 

Sun's  mean  horizontal  parallax,  Arts.  15,  21. 

Sun  and  moon's  equatorial  horizontal  parallaxes,  Art.  7. 

Moon's  parallax  reduced  to  latitude  45°  (Lunar  Eclipses),  Art.  189. 

Sun's  mean  semidiameter,  Art.  21. 

Sun  and  moon's  true  semidiameters,  Art.  21. 

Ratio  of  moon's  semidiameter  s,  to  that  of  the  earth  TT,  Section  XXI. 

Distance  between  the  sun  and  moon,  Art.  25. 

Eatio  of  moon's  distance  r  to  that  of  the  sun  rf,  Art.  25. 

—  1  —  6,  Art.  25. 

Inclination  of  the  moon's  orbit  to  the  ecliptic,  Art.  8. 

Sidereal  Time,  Art.  27. 

Hour  angle  of  the  point  Z  at  the  Greenwich  Meridian,  Art.  27. 

Its  hourly  variation. 

Mean  time  hours  reduced  to  sidereal  time,  Art.  27. 

Coordinates  of  the  centre  of  the  shadow,  Art.  28. 

Mean  hourly  variations,  Art.  30. 

Absolute  hourly  variations,  Art.  30. 

y  transformed  to  take  account  of  the  earth's  spheroid,  Art.  33. 

Eadii  of  the  penumbra  and  umbra  on  fundamental  plane,  Art.  31. 

Eadius  on  the  earth's  surface,  Arts.  116,  150. 

Also  radius  of  the  earth's  shadow  in  Lunar  eclipses,  Art.  189. 

==•  pf  cos  d.    Quantities  in  the  Eclipse  Tables,  Arts.  27,  34. 

Angles  of  the  cones  of  shadow,.  Art.  31. 

i  =  tan/,  t,  =  tan/!  as  used  in  the  formulae,  Art.  31. 

Distances  of  the  vertices  of  cones  above  fundamental  plane,  Art.  31. 

Constants  for  cones,  Art.  31. 

Auxiliaries  in  the  General  Formulae,  Arts.  81,  86. 


*  These  symbols  or  others  like  them  are  found  in  other  parts  of  this  list. 

ix 


PRINCIPAL  NOTATION. 


i'c' 


e 

E 

e 

M 

M 

m 

Z 

N 

n 

Q 


Quantities  in  Eclipse  Tables,  Art.  32. 

Their  hourly  variations,  Art.  32. 

Radius  of  the  earth  in  the  fundamental  plane,  Arts.  50. 

The  earth's  radius  for  any  place,  Arts.  42,  149. 

Transformation  of  p,  Arts.  32,  42. 

Angle  of  the  resultant  of  the  motion  of  the  shadow  and  a  point  on  the 

earth's  surface,  Arts.  32,  44,  70,  97. 
The  distance  at  each  hour,  Art.  32,  etc. 

Equation  of  Time  (chiefly  used  in  Lunar  Eclipse),  Arts.  163,  189. 
Eccentricity  of  the  earth's  spheroid,  Art.  20. 
Magnitude  of  an  eclipse,  Solar,  Arts.  79,  156 ;  Lunar,  Art.  191. 
Angle  of  positions  of  any  point  of  centre  line  with  axis  F,  Arts.  58,  79. 
Its  distance  on  the  fundamental  plane,  Arts.  58,  79. 
Zenith,  The  axis  of  Z,  Pole  of  fundamental  plane,  Art.  38. 
Angle  of  the  path  of  shadow  with  axis  of  Y,  Arts.  95-7. 
Motion  of  shadow  in  one  hour,  Arts.  95-7. 
Angle  of  position  of  centre  of  shadow  from  any  point  on  the  cone  of 

shadow,  Arts.  80,  94,  97,  118,  137. 
Geographical  latitude  of  any  place,  Art.  149. 
Geocentric  latitude  of  the  same  place,  Art.  149. 
Latitude  to  take  account  of  the  earth's  spheroid,  Art.  42. 
Coordinates  of  any  place  on  the  earth's  surface,  Art.  85. 
Their  hourly  variations. 
Sun's  zenith  distance,  Section  XX. 
The  coordinate  d  d  —  cos  (3,  Arts.  81,  85. 
Transformation  for  the  earth's  spheroid,  Art.  85. 
Angles  of  position  on  fundamental  plane,  Art.  50. 
Angle  in  the  Extreme  times  explained  in  Art.  58. 
Another  angle  in  Maximum  curve  defined,  Arts.  72  to  75. 
Parallactic  angle,  Art.  87. 
Altitude  of  the  sun  above  the  horizon,  Art.  87. 
Height  of  a  place  above  fundamental  plane,  Art.  87. 
Hour  angle  of  any  place,  and  Local  Apparent  Time  Arts.  50, 108,  109, 

157. 

West  Longitude  of  any  place  «  =  //,  —  $,  Art.  50. 
Epoch  Hour,  Art.  30,  or  assumed  time. 
Computed  correction  for  assumed  time  T0. 
The  time  of  any  phenomenon  T=  T0  -}-  r. 
Magnitude  or  Degree  of  obscuration,  Arts.  79,  189,  191. 
Sum  of  any  two  or  more  quantities,  Art.  30. 
Used  to  denote  a  distance,  Art.  68. 
Finite  differences  of  computed  quantities,  Arts.  18,  29. 
Symbol  of  differentiation,  Arts.  28,  182. 
Angle  of  position  from  vertex  of  sun  oi*  moon,  Art.  153. 
Notation  in  Addition  and  Subtraction  logarithms,  Arts.  28,  35. 


LIST  OF  PLATES  AND  FIGURES. 

WITH  INDEX  TO  GEOMETRICAL  CONSTRUCTIONS. 


PLATE  FIG.  PAGE 

1.    Kelation  of  the  Conjunction  Points  to  the  Moon's  Node  ....     31 

I.  2.     Total  Eclipse,  1904,  September  9 54 

II.  3.    Total  Eclipse,  1860,  July  18  (CHAUVENET'S  Example)     ...  57,  60 

Quantities  in  the  Eclipse  Tables  explained 54  to  65 

Principles  of  Spherical  Projection 56 

The  angle  E  explained 60 

Motion  of  the  Eclipse  Series 60 

Central  Eclipse  at  Noon 121 

The  extreme  times  and  interior  contacts 73-6 

Extreme  Times  of  Central  Eclipse  116 ;  curve 119 

III.  4.     The  Umbral  Cone  explained 62 

5.  Equation  78  explained.     The  four  contacts 70 

6.  The  Kising  and  Setting  Curve 78 

IV.  7.  Peculiarities  of  Eclipses  and  their  Curves 81,  140 

V.  8.  Maximum  curve  and  extreme  times  of  Limits.   The  office  of  the 

angle  E  in  this  curve 84-7 

Limits  for  the  angle  Q  in  Outline  curves 95 

Curves  for  Outline  92.     Equation  130  explained  for  £ 96-7 

9.     The  formation  of  the  Node  or  Multiple  Point 89 

10.  Greatest  Eclipse,  90.     Magnitude 89,  90 

11.  The  Angle  Q  generally  explained 92 

12.  Geographical  Positions  in  Outline  Curves 96-8 

13.  Dr.  HILL'S  Formulae  explained 99 

14.  Limits  f or  Q;  N.  and  S.  limiting  curves 108 

15.  Eelations  of  the  angles  Q  and  E 110 

16.  Check  formula  for  limits  of  Umbra  explained 131 

VI.         17.     Chart  Total  Eclipse,  1904,  September  9 141 

VII.         18.  Chart  Total  Eclipse,  1886,  August  8  .       141 

19.  The  reverse  curvature  of  Outline  Curves 144 

20.  Prediction.     Explanation  of  check  formulae 154 

VIII.         21.  Method  by  Semidiameters.  Projection 158 

22.  Method  by  Semidiameters.     Explanation  of  Formulae  ....   163-4 

IX.         23.  Shape  of  the  Shadow  upon  the  Earth.     The  Ellipse 179 

24.  Circular  Shadow 187-8 

25.  The  constant  k— Effect  of  Lunar  mountains ]  90 

26.  The  earth  considered  smaller  in  the  formulae  for  Eclipses  than  in 

Occultations 194 

X.         27.     Lunar  Eclipses.     Projection,  1902,  April  22 211-14 

XI.      i  ^'    Occultations,  Projection,  Explanation  of  Formulae 231 

1 29.     Limiting  Parallels  graphically  explained 233 

249 


THE 

THEORY  OF  ECLIPSES. 


SECTION    I. 

INTRODUCTION. 

ARTICLE  1. — Retrospect. — There  are  two  methods,  radically  differ- 
ent from  each  other,  of  computing  a  solar  eclipse.  That  known  to 
the  earlier  astronomers  consists  in  finding  the  times  when  the  disks 
of  the  sun  and  moon  are  tangent  in  a  visual  line  from  the  observer ; 
or,  in  other  words,  when  the  centres  of  the  sun  and  moon  are  distant 
from  one  another  by  an  arc  in  the  celestial  sphere  equal  to  the  sum 
of  their  semidiameters. 

It  was  not  until  the  time  of  BESSEL  and  HANSEN  in  the  early  part 
of  the  nineteenth  century  that  this  subject  was  fully  developed,  and 
another  method  devised  for  computing  solar  eclipses,  by  considering 
the  cone  of  shadow  cast  by  the  moon,  and  its  passage  over  the  sur- 
face of  the  earth ;  and  similarly  for  lunar  eclipses  the  passage  of  the 
moon  through  the  earth's  shadow. 

In  solar  eclipses,  the  first  method  is  restricted  to  but  one  place  on 
the  earth's  surface  at  a  time ;  given  the  place,  the  times  may  be  found. 
The  second  method  is  the  reverse  of  this ;  given  the  time,  and  the 
place  where  the  phenomenon  is  seen  may  be  found.  Thence  by 
assuming  a  series  of  times,  all  places  on  the  earth's  surface  where 
the  same  phenomena  are  visible  may  be  made  known,  an  advantage 
which  the  first  method  does  not  possess.  The  position  of  the  cone 
of  shadow  is  determined  by  its  axis, — the  line  joining  the  centres 
of  the  sun  and  moon, — and  all  points  of  the  earth  and  shadow  are 
referred  to  a  Fundamental  Plane  taken  at  right  angles  to  the  axis, 
and  passing  through  the  centre  of  the  earth.  A  set  of  coordinate 
axes  may  now  be  assumed  at  pleasure.  BESSEL,  who  wrote  in 
1841-42,*  by  a  happy  thought  selected  the  intersection  of  this 

*  Astronomische  Untersuchungen,  2  vols.,  1841—42. 

17 


18  THEORY  OF  ECLIPSES.  1 

plane  with  the  earth's  equator  as  the  axis  of  X.  HANSEN,  who 
followed  him  in  1858,*  assumed  the  intersection  of  this  plane  with 
the  ecliptic  as  the  axis  of  X.  The  former  method  has  many  advan- 
tages over  the  latter.  "  As  a  refined  and  exhaustive  disquisition  upon 
the  whole  theory,"  says  Professor  CHAUVENET,f  "  BESSEL'S  Analyse 
der  FinsternissCj  in  his  Astronomische  Untersuchungen,  stands  alone." 

Professor  CHATJVENET  has  generally  followed  BESSEL'S  method  ; 
but  in  many  of  the  problems  has  given  his  own  solutions,  making 
his  work  the  best  yet  published  on  the  subject  of  eclipses. 

While  Astronomy  was  advancing  in  this  direction,  other  branches 
of  science  were  no  less  active  in  lines  that  were  destined  eventually 
to  have  great  influence  upon  Astronomy.  As  early  as  1814,  though 
they  had  been  observed  previously,  FRAUNHOFER  studied  the  dark 
lines  in  the  solar  spectrum  to  which  his  name  has  been  given ;  but 
it  was  not  until  1859,  when  KIRCHOFF  made  his  great  discovery  of 
the  reversal  of  the  spectrum,  that  their  nature  and  origin  became 
known,  and  Spectrum  Analysis  burst  into  a  science,  the  connecting 
link  between  Chemistry  and  Astronomy. 

In  the  arts  also  Photography  was  progressing,  and  in  1860  was 
first  made  use  of  in  observing  a  solar  eclipse.  It  was  at  once  seen 
that  during  a  total  eclipse  spectrum  analysis  could  advantageously 
be  used  for  observation,  but  it  was  not  until  1868  and  1869  that 
an  opportunity  occurred  for  this  purpose.  Total  solar  eclipses  then 
assumed  a  much  greater  importance  than  heretofore.  Hitherto  they 
had  been  observed,  simply  by  the  times,  for  the  correction  of  the 
ephemeredes  ;  now  the  chemical  constituents  of  the  sun  can  be  deter- 
mined, and  the  corona  and  red  protuberances  observed  understand- 
ingly.  Then  to  meet  the  new  wants  of  astronomers,  when  our  Nau- 
tical Almanac  was  enlarged  in  1882  by  Professor  NEWCOMB,  the 
data  for  eclipses  were  much  improved  ;  BESSEL'S  tables  and  formulae 
substituted  for  SAFFORD'S  transformation  ;  the  path  of  totality  given 
by  the  latitude  and  longitude  of  consecutive  points ;  the  duration  of 
totality  on  the  centre  line  given ;  the  charts  enlarged  and  improved, 
and  quantities  referred  to  the  Greenwich  meridian  instead  of  that  of 
Washington.  This  brings  the  subjects  down  to  the  present  time. 

2.  General  Remarks  on  the  Computation. — This  should  be  made  for 
Greenwich  mean  time  and  the  Greenwich  meridian,  for  which  all  the 
quantities  required  are  given,  both  in  the  English  and  American 

*  Theorie  de  F  Eclipse  du  soleile  et  des  phenomenes  que  s'y  rattachent,  Leipsic,  1858. 
t  Spherical  and  Practical  Astronomy,  1863. 


2  INTRODUCTION.  19 

Nautical  Almanacs.  If  desired,  all  the  results  can  be  subsequently 
reduced  to  the  local  time  and  the  meridian  of  any  place.  Although 
the  present  work  is  complete  in  itself,  yet  on  account  of  the  numer- 
ous references  to  CHAUVENET'S  Chapter  on  Eclipses,  the  reader  will 
find  it  to  his  advantage  to  have  a  copy  of  that  work  at  hand,*  in 
which  also  may  be  found  the  derivation  of  the  several  formulae  here 
quoted.  For  the  numerical  computation  the  formulae  and  precepts 
in  the  present  volume  will  be  found  all-sufficient. 

It  is  recommended  that  the  computer  take  up  the  parts  of  the 
work  in  the  order  here  given  by  sections,  on  account  of  quantities 
in  a  preceding  section  being  often  required  in  those  following.  For 
the  same  reason  the  formulae  in  one  section  are  given  in  the  order 
most  convenient  for  use,  followed  by  precepts  for  the  computation. 

The  graphic  method  of  representing  quantities  in  the  formulae, 
which  thus  explains  the  whole  theory  of  eclipses,  has  not,  so  far  as 
the  author  is  aware,  been  made  use  of  in  explaining  an  intricate 
series  of  equations,  such  as  those  for  the  principal  times,  the  central 
line,  outline,  etc. 

The  computer  should  provide  himself  with  good  tables.  BHUHN'S 
seven-place  logarithms  are  the  best,  on  account  of  the  first  six 
degrees  being  given  to  every  second ;  without  this  it  will  be  diffi- 
cult to  get  the  sines  of  the  numerous  small  angles.  ZECH'S  Addition 
and  Subtraction  Logarithms  (seven-place)  are  convenient,  but  as  they 
are  used  only  a  few  times,  they  may  be  dispensed  with.  GAUSS'  five- 
place  logarithms  given  to  seconds,  or  NEWCOMB'S  five-place  loga- 
rithms given  to  decimals  of  a  minute,  may  be  used  for  the  greater 
part  of  the  computation  ;  and  when  using  four-place  logarithms,  they 
will  be  found  much  more  convenient  than  the  four-place  tables,  on 
account  of  there  being  no  interpolations  necessary.  All  operations 
are  algebraic,  and  strict  regard  must  be  paid  to  the  sign  before  a  term, 
as  well  as  to  the  sign  of  the  term  itself,  which  latter  is  dependent 
upon  the  signs  of  its  factors.  A  beginner  will  find  that  many  of 
his  mistakes  arise  from  disregard  to  these  rules  of  algebra. 

In  such  a  computation  as  the  eclipses,  where  signs  change  fre- 
quently, it  is  the  author's  custom  to  write  the  signs  before  each 
logarithm,  whether  the  number  be  plus  or  minus ;  or  before  the  first 
and  last  of  each  group.  By  so  doing  the  omission  of  a  sign  is  at 
once  perceived.  The  sign  may  be  omitted  before  a  log.  tangent,  as 
the  quadrant  is  determined  by  the  signs  before  its  sine  and  cosine. 

*  Manual  of  Spherical  and  Practical  Astronomy,  by  WILLIAM  CHAUVENET,  Phila- 
delphia, J.  B.  Lippincott  &  Co.,  1863. 


20  THEORY   OF   ECLIPSES.  2 

These  computations,  especially  those  for  prediction,  should  be  made 
with  the  greatest  accuracy.  They  serve  not  only  to  warn  the  astrono- 
mer when  to  watch  for  his  observed  times,  but  also,  when  compared 
with  the  observation,  serve  to  indicate  the  accuracy  of  the  tables, 
which  is  of  far  greater  importance. 

No  long  computation  can  generally  be  made  except  upon  paper 
ruled  so  as  to  keep  the  work  in  lines  and  columns.  In  the  Nautical 
Almanac  office  several  kinds  are  provided.  That  mostly  used  is  in 
sheets  of  the  best  stout  linen  paper,  21  inches  broad  by  16  inches, 
ruled  lengthwise  into  spaces  about  nine  to  the  inch,  and  vertically 
eleven  to  two  inches,  which  gives  85  lines  to  a  page.  One  figure  is 
usually  placed  in  a  space  so  formed.  A  solar  eclipse  will  generally 
require  three  such  sheets,  and  a  lunar  eclipse  one  side  of  a  sheet. 

Some  of  my  readers  may  possibly  charge  me  with  having  been  too 
minute  in  the  explanations  and  in  giving  the  examples  in  full.  The 
latter  is  the  suggestion  of  a  student  in  this  work  who  wrote  to  me 
for  explanations.  An  example  given  in  full  will  generally  show  how 
each  result  is  obtained ;  but  results  only,  as  examples  are  generally 
given,  often  fail  to  explain  the  difficulties  a  student  meets  with. 

3.  Differentiated  Quantities. — In  the  matter  of  the  graphic  repre- 
sentation of  a  formula  as  made  use  of  in  this  work,  the  author  is  of 
opinion  that,  except  in  very  simple  cases,  formula?  derived  from  dif- 
ferentiation cannot  always  be  shown  graphically  in  the  same  manner 
as  the  primitive  equation ;  especially  after  they  have  been  much 
transformed  or  simplified.  The  reasons  why  this  cannot  be  done 
are  various.  Differentiated  quantities  are  usually  so  small  that  they 
cannot  be  shown  upon  the  same  scale  as  the  terms  which  compose 
them.  Another  reason  may  be  that  the  equation  has  various  values, 
depending  upon  the  value  of  differential  of  the  independent  variable, 
and  this  value  may  not  at  once  be  evident ;  that  is,  we  may  have  its 
value  in  figures,  but  cannot  use  them  in  a  strictly  graphical  man- 
ner. To  illustrate  this,  take  the  following  example  from  the  eclipse 
formulae : 

x  =  r  cos  d  sin  (a — a). 
Differentiate 

dx  =  r  cos  d  cos  (a  —  a)  d(a  — a) 
=  0.001018  r  cos  (a  —  a). 

d(a  —  a)  in  CHAUVENET'S  example  is  about  35  minutes  of  arc,  which 
in  parts  of  radius  gives  the  decimal  in  the  third  equation.  Now  we 
can  easily  show  x  graphically  from  its  factors.  We  can  also  show 


3  INTRODUCTION.  21 

the  factors  r  cos  d  cos  (a  —  a),  either  separately  or  combined  ;  hence, 
dx  must  be  about  one-hundredth  part  of  this  latter.  But  this  is  not 
strictly  a  graphical  representation,  but  numerical,  and  we  do  not  show 
dx  "  in  the  same  manner  "  as  we  do  x.  In  this  equation,  r  has  a  value 
of  about  60  units ;  this  and  the  small  numerical  term  cannot  possibly 
be  shown  on  the  same  scale  as  above  stated. 

Where  a  differentiated  equation  has  been  transformed,  the  quantities 
become  so  estranged,  as  it  were,  that  it  is  nearly  impossible  to  follow 
them  ;  nevertheless  some  expedient  may  generally  be  found  to  illus- 
trate the  required  differential.  In  this  manner  we  can  show  dx ;  it 
is  simply  the  difference  between  two  values  of  x  computed  for  two 
successive  hours,  or  the  hourly  motion  of  x;  and  it  is  shown  in 
several  of  the  subsequent  figures  when  successive  values  of  x  are 
given. 

4.  The  Darkness  at  the  Crucifixion. — Before  closing  this  section, 
the  author  may  be  pardoned  for  a  few  words  upon  the  arguments 
of  certain  atheists  and  others  who  deny  the  miraculous  darkness  at 
the  crucifixion,  stating  that  it  was  caused  by  a  total  eclipse  of  the 
sun.  But  this  could  not  possibly  have  been  the  cause  of  the  dark- 
ness. The  Jewish  months  were  lunar,  and  commenced  at  new  moon. 
Abib  was  the  first  month ;  preparations  were  made  on  the  tenth  day, 
and  the  Passover  eaten  on  the  fourteenth  day  (Ex.  xii.  2,  6 ;  Deut. 
xvi.  1).  The  Crucifixion  took  place  at  the  time  of  the  Passover, 
when  the  moon  was  full,  now  commonly  called  the  Paschal  Full 
Moon ;  and  the  darkness  was  not  caused  by  an  eclipse  of  the  sun, 
which  can  take  place  only  at  new  moon. 

The  author,  having  computed  the  eclipses  yearly  for  the  Nautical 
Almanac  for  twenty-three  years,  a  longer  period  than  that  of  any  of 
his  predecessors,  resigned  that  portion  of  his  work  after  completing 
the  computation  of  the  eclipses  for  1905. 


UNlV 
PAET  I. 

SOLAR  ECLIPSES. 


SECTION    II. 

THE  CRITERION. 

5.  AT  the  very  threshold  of  this  subject  we  meet  with  an  article 
without  which  CHAUVENET'S  chapter  would  be  incomplete,  and  yet 
one  which  the  practical  computer  will  seldom  use — an  evidence  of 
the  completeness  of  his  work. 

The  object  of  this  section  is  merely  to  ascertain  whether  a  solar 
eclipse  will  occur  at  or  near  a  given  time.  Eclipses  are  not  scattered 
irregularly  through  the  year,  but  can  occur  only  when  the  sun  is  near 
one  of  the  moon's  nodes.  These  have  been  styled  by  Professor  NEW- 
COMB  as  eclipse  seasons,  the  middle  of  which  are  the  instants  the  sun 
passes  the  longitude  of  the  moon's  node ;  the  seasons  are  therefore 
about  six  months  apart,  and  a  solar  eclipse  can  occur  only  when  the 
sun  is  within  18  days  of  the  node,  before  or  after;  and  a  lunar 
eclipse  11|  days  before  or  after.* 

This  precept  is  the  most  convenient  we  have  for  commencing  our 
search,  for  which  we  will  take  the  year  1902.  Towards  the  close  of 
Part  I.  of  the  Nautical  Almanac,  which  is  given  for  the  Greenwich 
meridian,  page  284  for  this  year,  headed  MOON,  we  find,  "  &  Mean 
Longitude  of  Moon's  Ascending  Node."  It  will  be  noticed  that  it 
has  a  retrograde  motion  of  about  20°  in  a  year.  The  descending 
node  £3  differs  180°  from  the  values  here  given.  Now,  comparing 
this  with  the  Sun's  Longitude,  on  page  III.,  for  each  month,  we  find 
that  they  are  the  same  on  the  following  dates : 

Page  58,  1902,  April  25,  I'  =  V  =  214°  —  180°  =  34° 
"   166,  Oct.    19,  I'  =  £  =  205 

These  dates  are  the  middle  of  the  eclipse  seasons,  and  the  seasons 
themselves  are  from  April  7  to  May  13,  and  Oct.  1  to  Nov.  6. 

We  are  now  prepared  to  use  CHAUVENET'S  Criterion,  from  which 
we  have,  in  addition  to  the  general  notation  in  this  work,  the  following : 

*  Popular  Astronomy,  pages  29,  30. 

23 


24  THEORY   OF   ECLIPSES.  6 

/'  Sun's  true  longitude. 

I     fi         Moon's  true  longitude  and  latitude. 

Alf  Al  dp  Motions  in  12  hours. 

t  Interval  of  time  to  conjunction. 

6.  If,  at  the  time  of  conjunction  in  longitude,  we  find,  regarding 
the  moon's  latitude,  — 

/5  <  1°  24'  34",  Eclipse  is  certain  * 

/?  >  1    34  47,  Eclipse  is  impossible,  (1) 

Between  these  limits,  Eclipse  is  doubtful. 

If  the  eclipse  is  very  doubtful,  by  the  value  approaching  the  larger 
limit,  we  may  then  use  the  following  more  accurate  formula  : 

-.Tr'  +  s-f-g'-f  25",  Eclipse  certain.  (2) 


We  will  pass  over  the  ascending  node  and  examine  the  other  one. 
On  pages  280-83  of  the  Nautical  Almanac  we  have  the  moon's  lati- 
tude and  longitude,  and  on  page  58  these  quantities  for  the  sun. 
We  need  only  look  near  these  dates  when  the  moon's  latitude  is  near 
0.  We  therefore  have  the  dates  April  9  and  May  6  to  examine  for 
solar  eclipses  and  April  23  for  a  lunar.  For  this  latter  the  sun's 
longitude  will  differ  180°  from  that  of  the  moon,  and  we  will  neg- 
lect this  until  we  take  up  Lunar  Eclipses.  From  these  pages  we 
find  for  1902  May  7  12*.  . 

For  the  sun,     I1  =  46°  28' 

For  the  moon,  I  =  47    12  p  =  -  1°  10'  32.7" 

We  see  by  inspection,  without  doing  anything  further,  that  there  is 
an  eclipse  at  this  date.  Passing  to  the  other  date,  April  23,  we  take 
from  the  Almanac  to  the  nearest  minute  as  follows  : 


Date. 
April  8  . 
"   8.5  . 
"  9  . 
"  9.5  . 

V. 

.  17° 
.  18 
.  18 
.  19 

44' 
13 
42 
12 

+  29 
29 
+  30 

I. 
16° 
23 
31 
38 

41' 

59 
20 
42 

+  7  18 
7  21 

+  7  22 

I—  I. 
—  1°  3' 
+  5  46 
12  38 
+  19  30 

+  6 
6 

+  6 

49 
52 
52 

/ 
+  1° 
0 
+  0 
—  0 

1-39 

17   40 

11-^ 

The  sun's  longitude  is  here  interpolated  to  12  hours,  and  the  differ- 
ence I  —  V  gotten  in  the  third  group.  Now  when  the  sun  and  moon 
are  in  conjunction,  the  difference  I—I'  must  be  zero.  If  t  is  the 
interval  from  the  first  date,  April  8,  to  the  time  of  conjunction,  and 
AU  Al  the  motion  between  successive  dates,  we  must  have 

V  +  tAV=l  +  t  AL 

*  I  have  slightly  changed  CHATJVENET'S  values  here  for  a  reason  given  on  a  subse- 
quent page  of  this  section. 


6  THE  CRITERION.  25 

Whence 


which  is  the  usual  formula  for  interpolation,  omitting  second  differ- 
ences.    In  the  present  case  we  have 

170  44/  _  16r  41//       1°  3/ 

t  =  •  --  =  ~  -  •  —  -  =  T  U.1D. 

7°  18'  —  29'  6°  49' 

We  may  check  this  time  as  follows  to  ascertain  if  the  longitudes  are 

the  same  : 

For  the  sun,     17°  44'  +  .15(29)       =  17°  48', 
For  the  moon,  16    41  +  .15(7°  18)  =  17    47', 

which  is  quite  near  enough  for  our  purpose. 
Now  interpolating  /9  for  this  interval,  we  have 

0=  +  l°36'-6'  =  +1°30'. 

Comparing  this  with  the  Criterion  equation  (1),  we  find  the  lati- 
tude is  between  the  limits,  and  rather  doubtful;  so  we  will  now  look 
to  equation  (2). 

7.  The  above  time  0.15  is  a  fraction  of  12A,  and  in  hours  is  1.80,, 
which  is  the  time  of  New  Moon,  given  in  the  Almanac  on  page 
XII.  for  each  month.  It  agrees  very  well  with  the  time  given 
there,  lh  50m.l.  For  this  time  we  must  interpolate  the  several 
quantities.  The  sun's  parallax  is  given  on  page  285,  the  semi- 
diameter  on  page  L,  for  each  month  ;  and  the  moon's  parallax  and 
semidiameter  on  page  IV.  for  each  month.  The  semidiameters  given 
in  the  Almanac  are  marked  apparent  semidiameters,  because  they 
are  affected  by  a  constant  of  irradiation  ;  this  must  be  numerically 
deducted  after  interpolation.  It  has  been  changed  sometimes,  but 
its  value  may  be  found  in  the  Appendix  of  the  Almanac.  Under 
the  elements  of  the  eclipses  these  quantities  are  called  the  true  semi- 
diameters  because  they  are  so.  Interpolating  from  the  above  pages, 
we  find 

Moon's  parallax,                 TT  60'   I'M 

Sun's  parallax,                     TT'  8  .8 

TT-TT'  59   52   .3 

Sun's  true  semidiameter,     s'  15'  59".2  —  1.15  =  15  58  .1 

Moon's  true  semidiameter,  s  16  22  .5  —  1.50      16  21   .0 

Constant,  0  25  .0 

Sum  /?  =  1°  32'  36" 


26  THEORY  OF   ECLIPSES.  7 

We  seem  to  be  worse  off  than  we  were  before,  for  this  value,  which 
is  more  nearly  accurate  than  the  previous,  being  nearer  to  the  outer 
limit,  is  more  doubtful.  Recourse  must  now  be  had  to  the  rigorous 
method;  but  we  will  not  pursue  this  example  further.  There  is,  in 
fact,  a  small  partial  eclipse  at  the  north  pole,  and  it  may  interest  the 
reader  to  know  how  it  has  diminished  during  the  past  fifty  years. 
The  magnitudes  are  as  follows  : 

1848,  March  6,  magnitude  0.269  _  ^ 
1866,  "  16,  "  0.216  _?5 
1884,  "  26,  "  0.141  -6 
1902,  April  8,  "  0.065  __  76 
1920,  "  18,  "  —0.011 
Sun's  diameter  1,000 

The  last  value  here  in  1920  is  assumed  from  the  differences  to 
accord  with  those  above,  and  the  negative  sign  of  0.011  shows  that 
there  will  be  no  eclipse,  this  being  the  distance  between  the  moon's 
shadow  and  the  earth.  The  quantity  is  only  approximate. 

Should  the  reader  desire  to  compare  the  time  of  conjunction  com- 
puted above  with  that  given  in  the  Almanac  for  this  eclipse,  he  must 
remember  that  we  computed  the  conjunction  in  longitude,  whereas 
the  Almanac  gives  it  in  right  ascension  ;  these  times  are  not  the 
same,  and  the  semidiameters  and  parallaxes  would  also  differ ;  more- 
over, the  above  values  are  not  rigorously  interpolated. 

8.  Rigorous  Formulae. — No  example  of  this  will  be  here  given,  as  it 
will  probably  be  seldom  required.  The  preliminary  work  is  as  follows : 

Take  from  the  Nautical  Almanac  the  data  as  we  did  in  the  pre- 
vious example,  but  retaining  all  the  decimals.  Copy  the  moon's 
longitude  and  latitude  to  include  at  least  two  dates  beyond  the  times 
of  conjunction  and  the  time  when  the  latitude  is  zero.  Copy  the 
sun's  longitude  for  at  least  three  dates  each  side  of  conjunction ;  that 
is,  three  days,  interpolating  carefully  to  12  hours.  Difference  these 
and  form  the  column  I —  /'  as  before.  Compute  the  time  of  conjunc- 
tion as  before,  but  using  the  formula  given  in  the  next  section  under 
the  Elements,  and  used  there  for  conjunction  in  right  ascension.  Sec- 
ond differences  must  be  used  here  throughout,  for  which  Tables  II. 
and  III.  may  be  used,  and  at  least  one  decimal  of  a  second  retained. 
This  time  of  conjunction  may  be  checked  by  interpolating  the  two 
longitudes,  and  they  should  agree  exactly.  Also  interpolate  the  lati- 
tude exactly,  using  five-place  logarithms,  for  the  terms  depending 
upon  first  differences. 


8  THE  CRITERION.  27 

We  next  have  the  following  formulae  and  notation  : 

tan/9  tan/1/? 

tan    I  = = ( 4 ) 

Bin(a~/)        sinJJ 

tan    I'=— —  tan/  (5) 

A   •""•    J. 

^cosPO  — *'  +  «  +  «'  (6) 

/  Inclination  of  the  moon's  orbit  to  the  ecliptic. 
I'  An  auxiliary  angle. 

A    The  quotient  of  the  moon's  motion  in  longitude  divided  by  that  of 
the  sun. 

9.  CHATJVENET  does  not  show  how  to  get  the  inclination  J,  nor  is 
it  given  in  the  Almanac;  but  it  is  found  as  above  by  the  simple  right- 
angled  triangle  between  the  moon  and  the  node,  either  before  or  after 
its  passage.  The  longitude  of  the  node  is  the  same  as  the  moon's 
longitude  at  the  instant  when  the  latitude  is  zero.  It  is  given  in  the 
Almanac  to  the  tenth  of  a  minute,  but  as  it  may  be  required  closer 
than  that,  it  can  be  computed.  It  will  be  a  little  more  convenient, 
instead  of  computing  the  node  and  then  the  term  Q,~19  to  use  the 
second  form,  in  which  Al  and  Aft  are  the  differences  for  12  hours, 
just  at  the  node. 

For  A  the  differences  at  the  time  of  conjunction  must  be  used, 
computing  by  logarithms  to  five  places.  With  ZECH'S  Logarithms, 
the  subtraction  table  with  the  argument  log  ^,  gives  at  once  log 

>  and  thence  tan  P.     Cos  ft  is  already  computed,  hence  the 

A  ""~"~  J. 

first  member  ft  cos  P. 

Then  from  the  Almanac  take  out  the  moon's  semidiameter  and 
parallax,  page  IV.,  for  the  month ;  the  sun's  semidiameter,  page  I., 
and  the  parallax  at  the  close  of  Part  I.,  page  285,  or  thereabouts ; 
all  of  which  must  be  interpolated  carefully  to  the  time  of  conjunc- 
tion, using  second  differences  where  necessary.  The  semidiameters 
must  next  be  corrected  by  deducting  the  constants  of  irradiation, 
which  have  been  changed  at  times,  but  which  are  given  in  the 
appendix  of  the  Nautical  Almanac.  Those  that  have  heretofore 
been  used  are  given  in  the  next  section,  under  the  head  of  Con- 
stants. The  second  member  of  equation  (6)  may  now  be  formed, 
and  the  comparison  made ;  if  the  first  member  is  the  least  numeri- 
cally, an  eclipse  will  occur. 

I  would  hardly  recommend  this  method,  except  on  rare  occasions. 


28 


THEORY  OF   ECLIPSES. 


Generally  there  are  other  methods  of  discovering  an  eclipse,  but  if 
they  cannot  be  used,  this  method  is  lost  work  if  there  is  an  eclipse ; 
so  it  might  be  better  to  suppose  there  to  be  one,  and  proceed  by  the 
regular  formulae  given  in  a  subsequent  section,  computing  the  eclipse 
tables  for  only  three  hours  as  far  as  I  for  penumbra.  Then  if  the 
first  approximation  discloses  an  eclipse,  compute  the  tables  for  other 
hours  that  may  be  required,  and  proceed  in  the  regular  way.  There 
will  then  be  no  loss  of  the  work  done. 


10.  The  Semidiameters  and  Parallaxes. — The  values  of  these  given 
by  CHAUVENET,  on  page  438,  seem  to  need  revision,  the  moon's 
least  parallax  being  in  error  by  at  least  one  minute.  I  found  that 
authorities  are  not  agreed  as  to  the  moon,  and  then  consulted  Pro- 
fessor RUEL  KEITH,  who  has  computed  the  moon's  semidiameter 
and  parallax  for  the  Nautical  Almanac  for  thirty  consecutive  years  ; 
and  his  reply  was,  that  "  the  moon  changes  its  distance  so  irregularly 
it  is  hard  to  follow  it  by  rule."  My  only  recourse,  and  I  believe 
it  is  the  best  authority  yet,  is  to  take  the  extreme  values  of  the 
moon's  parallax  from  all  the  eclipses  I  have  computed  from  1883  to 
1905,  both  years  inclusive.  This  I  have  in  a  bound  volume,  giving 
the  full  data  of  all  these  eclipses.  These  extremes  may  not  be  the 
greatest  possible,  but  they  are  at  least  better  than  those  heretofore 
given  as  correct,  for  the  greatest  value  is  greater  than  that  given  by 
some  authorities,  and  the  least  value  is  less.  The  authorities  for  the 
Moon's  Parallax  are  as  follows : 


AUTHORITY. 

Greatest  Value. 

Least  Value. 

Mean  Value. 

CHAUVENET,  Astronomy,  1862  ~\  .  . 
WOOLHOUSE  (BARTLETT),  1836  C  .  . 
PEIRCE,  Trigonometry,  1852  J  .  . 
LARDNER,  Handbook,  Astronomy  .  . 
YOUNG  Astronomy 

61'  32." 

61   18. 
61   28 

52'  50." 

53  58. 
53  55 

57'  11." 

57  38. 
57     2 

PROCTOR  on  the  Moon  

61   28.8 

53  51.5 

57     27 

NEWCOMB  Astronomical  Constants  .  . 

57     255 

Eclipse  values  1  1883-1905  .... 
HANSEN'S  Tables  /  1880.  .  .  . 

61   27.3 

61  27.8 

53  55.9 

PEIRCE  wrote  in  1852  :  he  was  the  principal  mathematician  con- 
nected with  the  Nautical  Almanac  when  it  was  first  published — about 
that  time.  WOOLHOUSE'S  method  was  devised  for  the  English  Nau- 
tical Almanac  in  1836,  and  is  given  in  BARTLETT'S  Astronomy.  The 
eclipse  values  are  the  greatest  and  least  values  which  have  actually 


UNIVERSITY 


10 


THE  CRITERION. 


29 


occurred  at  the  time  of  any  eclipse  during  the  years  1883  to  1905 
inclusive.  It  seems  as  if  WOOLHOUSE,  the  earliest  writer,  had 
decreased  the  least  value  and  increased  the  mean  so  as  to  make  it 
a  middle  value,  which  it  properly  is  not. 

The  value  61'  27".8  occurs  in  1880,  Dec.  31,  and  very  nearly 
in  1899,  Jan.  11;  and  53'  55".9  in  1901,  Nov.  11,  and  also  very 
nearly  this  value  in  1883,  Oct.  30.  I  have  not  had  occasion  to 
examine  other  portions  of  these  years,  and  cannot  say  that  these  are 
the  greatest  and  least  that  have  occurred,  but  they  are  so  during  the 
eclipse  seasons.  The  parallaxes  repeat  themselves  with  the  Saros 
from  which  I  found  the  value  61'  27 ".8.  Therefore,  assuming 
values  a  little  beyond  the  greatest  and  least,  and  with  NEWCOMB'S 
mean,  we  take  the  following  as  the  standard  values  for  the  moon  : 

Greatest,  61'  28".8.        Least,  53'  55".0.        Mean,  57'  2".52. 
And  the  semidiameters  are  gotten  by  the  formula : 

s  =  Tea  =  0.272274  *. 

For  the  sun,  taking  from  NEWCOMB'S  Tables  of  the  Sun  *  the  semi- 
diameter,  which  is  AUWER'S  value,  and  parallax,  we  have  the  fol- 
lowing : 

ADOPTED  VALUES. 


Greatest  Value. 

Least  Value. 

Mean  Value. 

Moon's  equa.  hor.  parallax  TT    .    .    .    . 
Sun's  equa  hor  parallax  TT'  

61'  28".8 
8     9 

53'  55".0 

8    .6 

57'    2".25f 
8    80 

Moon's  true  semidiameter  s  •    •    • 

16  44    4 

14  40    8 

15  31     79 

Sun's  true  semidiaineter  s' 

16  16    0 

15  43    8 

15  59    63 

And  from  these  we  obtain  the  values  given  above  in  formula  (1). 


/? 


24' 
34 


10".7  +  23" 

20  .6  +  26 


=  1°  24' 
.7  =  1    34 


34".5 
47   .3 


*  Astronomical  Papers,  prepared  for  the  use  of  the  American  Ephemeris  and  Nautical 
Almanac,  vol.  vii.,  part  i.  The  above  constant  of  the  sine  of  lunar  parallax  expressed 
in  arc,  57'  2x/.52,  is  also  used  in  these  tables,  p.  12. 

t  HANSEN'S  parallax  is  used  in  the  Nautical  Almanac.  This  value  of  sin  parallax 
is  given  by  Professor  NEWCOMB  in  his  Transformation  of  Hanserf  s  Lunar  Theory,  Astr. 
Papers,  vol.  i.,  p.  105,  sin  TT  57'  2".09.  The  arc  is  0".16  greater,  as  given  in  the 
text.  In  his  Astronomical  Constants,  p.  194,  Professor  NEWCOMB  has  deduced  the  value 
57'  2".52. 


30  THEORY   OF   ECLIPSES.  10 

The  mean  of  the  small  terms  is  almost  exactly  25",  as  CHAUVENET 
gives  it. 

11.  The  Saros. — This  term  is  derived  from  the  ancient  astronomers, 
but  is  now  applied  to  the  period  of  the  recurrence  of  eclipses.  It  is 
given  in  NEWCOMB'S  Popular  Astronomy  thus  : 

242  returns  of  the  moon  to  her  node    ....  6585.357  days 
19  returns  of  the  sun  to  moon's  node  ....  6585.780 

The  sun  and  moon  are  therefore  together  223  times  during  this 
period,  the  length  of  which  is,  according  to  Professor  NEWCOMB  in 
his  Recurrence  of  Solar  Eclipses,* 

6585.321222  days, 
or 

18" 10*  7*  42W  33«.6,  if  Feb.  29  intervenes  j  5  times'  j          (7) 

(  4  times.  ) 

In  searching  for  an  eclipse  for  the  years  1882  to  1899  we  must  use 
for  the  Saros  the  following : 


18*  £«  12»  51-  |  (7  Ha.) 

because  the  eclipses  before  1882  are  given  to  Washington  mean  time, 
and  this  value  gives  the  reduction  to  Greenwich  mean  time  from 
1882  to  1899  inclusive.  After  the  latter  date  the  value  in  equation 
(7)  should  be  used. 

Also,  as  stated  above,  there  being  19  conjunctions  of  the  sun  with 
the  same  node,  if  we  divide  the  above  number  by  19,  we  get  the 
interval  between  them,  346.59  days.  Hence,  the  sun  will  return  to 
the  node, 

365.25  -  346.59  =  18.66  days, 

earlier  each  year.f  These  are  mean  values,  and  may  vary  from  the 
actual  times,  but  perhaps  not  over  two  hours. 

It  is  a  remarkable  circumstance  connected  with  the  Saros  that  the 
longitude  of  the  moon's  perigee  is  nearly  the  same  after  18  years. 
The  sun  also  is  within  11°  of  its  former  place,  consequently  all  the 

*  Astronomical  Papers,  vol.  1.,  part  i.,  p.  17. 

f  Newcomb's  Popular  Astronomy,  1878,  p.  30,  has  a  misprint  giving  this  wrongly  as 
19f  days,  which  puzzled  me  when  I  first  took  up  the  study  of  eclipses. 


11  THE  CRITERION..  31 

quantities  affecting  an  eclipse — the  parallaxes,  semidiameters,  decli- 
nations, hourly  motions,  etc. — will  be  very  nearly  the  same  as  they 
were  18y  lld  previously;  and  the  eclipse  also  will  therefore  be  very 
similar. 

12.  If  the  sun  and  moon  start  out  together  at  a  node,  at  the  end 
of  the  Saros  they  will  be  together 

a  little  before  they  reach  the  node. 
And  if  this  is  the  ascending  node, 
as  in  Fig.  1,  the  moon  in  the  orbit  o 
will  consequently  be  a  little  below, 

or  south  of  the  sun  in  the  ecliptic  e,  \\         ^^^"^ ~ff o 

and  the  reverse  must  be  the  case  at 
the  descending  node ;  hence,  we 
have  a  valuable  and  easy  criterion  : 

At  Moon's  Ascending  Node,  series  is  moving  South,  )  ^ 

"          Descending    «          "  "        North.} 

This  holds  good  for  both  solar  arid  lunar  eclipses. 

The  moon's  nodes  can  be  found  from  the  Nautical  Almanac,  pages 
280-83,  or  thereabouts.  When  the  latitude  is  zero,  the  differences 
are  positive  at  the  ascending  node  and  negative  at  the  descending 
node. 

13.  We  will  take  the  year  1904  to  illustrate  the  period  of  the 
Saros.     By  reference  to  the  Nautical  Almanac  of  eighteen  years  pre- 
viously, we  find  under  the  "Eclipses  in  1886"  that  there  are  but 
two,  both  solar ;  selecting  the  August  eclipse  we  find 

Greenwich  mean  time  of  cf  in  R.  A.,  1886,  Aug.  29d  0*  59W 
And  the  Saros  as  above  (Feb.  29  4  times),     18y  11   7  42 

1904,  Sept.    9  8  41 

which  is  near  enough  to  the  true  time  for  us  to  take  out  our  data 
from  the  Almanac.  In  fact,  as  this  is  an  example,  we  may  say  that  it 
is  correct  within  10  minutes  of  the  correct  time,  which  is  closer  than 
it  generally  comes. 

14.  In  making  search  for  the  eclipses  for  a  given  year  the  best 
method  will  be :  first,  to  find  all  the  eclipses,  solar  and  lunar,  as 
given  by  the  Saros,  as  above.    Second,  to  ascertain  whether  any  have 
run  out.     The  example  given  above  and  the  comparison  of  magni- 


32  THEORY   OF   ECLIPSES.  14 

tudes  show  that  no  partial  solar  eclipse  whose  magnitude  is  as  much 
as  0.100  will  be  likely  to  run  out.  Sometimes  the  shadow  moves 
more  slowly.  In  case  of  very  small  magnitudes,  refer  back  36,  or 
preferably  also  54,  years,  and  very  likely  the  case  can  be  settled ;  or 
else  employ  the  formulae  (1)  or  (2).  Third,  to  find  whether  any  new 
series  enters.  This  is  more  difficult,  and  the  computer  must  be  abso- 
lutely certain  about  this  point.  If  he  possesses  access  to  OPPOLZER'S 
Canon  of  Eclipses  (in  German),  he  can  find  out  what  eclipses  will 
occur  in  the  future,  examining  especially  the  year  he  is  about  to 
compute  for.  It  would  be  well  also  to  look  18  years  ahead,  and  if  a 
new  series  is  found  to  enter,  then  it  would  not  be  safe  to  assume 
that  it  will  not  appear  during  the  year  he  is  searching  for,  because 
OPPOLZER'S  method  of  computing  is  approximate,  and  may  not 
have  caught  the  very  first  appearance  of  a  new  series.  If  the 
matter  is  still  doubtful,  or  the  computer  does  not  possess  a  copy  of 
OPPOLZER,  he  should  know  where  to  look  for  a  supposed  eclipse. 
Only  three  eclipses  can  possibly  occur  at  each  node,  two  solar  and 
one  lunar,  or  the  reverse :  one  large  eclipse  very  near  the  node  and 
two  partial  eclipses,  about  fourteen  days,  one  on  each  side  of  the  node. 
So  if  there  is  a  large  eclipse  at  the  node  and  one  partial,  fourteen  days 
distant,  there  may  be  another  partial  on  the  other  side.  It  some- 
times occurs  that  there  is  but  one  eclipse  at  each  node,  as  in  the  year 
1904.  In  this  case  a  lunar  may  enter  at  either  node.  This  occurs, 
according  to  OPPOLZER,  in  1958,  May  3.  If  there  are  two  eclipses, 
more  or  less  seven  days  from  the  node,  there  can  be  no  other.  For- 
mula (1)  or  (2)  should  be  used  in  these  cases. 

There  is  one  other  point  to  be  noted  :  if  the  eclipse  season  occurs 
in  December,  an  eclipse  may  occur  in  the  January  following,  and, 
vice  versa,  the  middle  of  the  season  occurring  in  January  of  the  next 
year  may  cause  an  eclipse  in  December  of  the  present  year. 

And  generally  under  very  doubtful  cases,  as  I  suggested  above,  I 
would  proceed  as  if  there  were  an  eclipse  rather  than  go  through  the 
rigorous  method  for  criterion.  In  this  case,  if  there  is  no  eclipse,  it 
will  show  itself  by  sin  <p  in  the  formulae  for  beginning  and  ending 
coming  out  greater  than  unity,  which  is  impossible,  and  the  computa- 
tion can  be  carried  no  farther. 


15  DATA  AND  ELEMENTS.  33 

SECTION  III. 
DATA  AND  ELEMENTS. 

15.  THE  data  for  a  solar  eclipse  are  required  for  six,  seven,  or 
eight  hours  of  Greenwich  mean  time,  according  to  the  size  of  the 
eclipse,  and  including  the  time  of  conjunction  in  right  ascension. 
We  will  proceed  as  follows  :  Taking  the  time  of  conjunction  as 
given  by  the  Saros  in  the  previous  section,  compare  the  sun  and 
moon's  right  ascensions,  interpolating  the  sun's  by  the  hourly  mo- 
tions to  seconds  only,  and  it  will  soon  be  found  between  what  hours 
the  conjunction  lies.  Then  with  these  values  of  the  sun  and  moon 
find  the  time  within  a  few  minutes  by  formula  (3)  used  above,  or 
by  that  used  for  the  elements  in  this  section.  This  gives  us  pretty 
nearly  the  time  of  conjunction  of  the  present  eclipse,  and  we  can 
now  find  out  how  many  hours  we  will  have  to  compute  for  in  the 
following  manner  : 

In  the  previous  section  we  found  the  time  of  an  eclipse  to  be  by 
the  Saros  1904,  September  9d  Sh  41m,  and  by  comparing  the  right 
ascensions  as  above,  we  find  a  more  correct  time  to  be,  we  will  sup- 
pose, Sh  47m.  Now  take  the  previous  eclipse  as  follows  : 


Eclipse  begins 
cf  in  R.  A., 
Eclipse  ends 

1886,  August. 

1886,  Aug.  28d  22*  Um  t 
«      29     0   58    ; 
"      29     3  32    ' 

140 
234 

1904,  September. 

9d    6*    lm 

841  'S 

11   15    234 

Apply  the  intervals  of  the  eclipse  of  1886  to  the  time  of  con- 
junction of  1904,  and  we  have  very  nearly  the  times  of  beginning 
and  ending,  so  that  we  will  have  to  compute  for  seven  hours  —  6  to 
12,  both  inclusive. 

Therefore,  copy  from  the  Nautical  Almanac  as  follows  : 

Sun's  K.  A.  and  Dec.  six  dates  from  p.  II.  for  the  month, 

log  radius  vector      "  "  III.       "  " 

semidiameter  "  "  I.       "  " 

Sidereal  time  one  date       "  II.      "          " 

Moon's  R.  A.  and  Dec.  for  the  eclipse  hours,  V.-XII.  " 

semidiameter,  six  dates  of  12  hours,      IV.       "  " 

16.  These  six  quantities  are  required  for  all  eclipses  for  interpola- 

tion.   The  sun's  parallax  can  be  taken  from  page  285  for  the  previous 

year,  if  this  year  is  not  in  print.    It  is  sensibly  constant  for  the  same 

day  of  each  year.    This  and  the  semidiameters  are  used  only  as  checks. 

3 


34  THEORY  OF  ECLIPSES.  16 

The  quantities  should  be  carefully  differenced,  and  then  interpolated 
by  the  differences  for  each  of  the  eclipse  hours  (the  semidiameters 
excepted).  They  should  be  interpolated  by  the  usual  formula  as 
follows,  applied  to  each  of  the  eclipse  hours  : 

a,=  a0  +  W,  +  '--)J1.  (9) 


The  sun's  right  ascension  should  be  carried  out  to  three  decimals  of 
time  and  the  declination  to  two  decimals  of  arc,  also  the  moon's 
parallax  to  three  decimals  of  arc  ;  log  r  depending  upon  it  will  then 
difference  smoothly,  which  is  a  great  advantage  ;  otherwise  it  will  not. 
The  interpolations  are  not  difficult,  but  require  care.  The  several 
values  of  the  second  term  of  the  formulae  when  applied  to  each  hour 
are  multiples  of  ^  of  the  first  differences  of  the  quantities  copied,  as 
above,  and  the  third  term  can  be  taken  from  Table  II.  of  this  work. 
This  table,  which  the  author  has  used  for  a  number  of  years,  is  in  two 
parts  —  for  right  ascension  in  time  and  for  declination  in  arc  ;  the  col- 
umns give  multiples  of  the  coefficient  depending  upon  the  argument  Av 
Log  r  can  be  taken  out  of  either  table,  considering  the  quantities  as 
whole  numbers  instead  of  as  decimals  in  the  margin,  and  moving  the 
decimal  points  in  the  body  of  the  table.  For  the  moon's  parallax, 
which  is  given  to  12  hours,  the  table  can  also  be  used,  considering 
that  2,  4,  6  hours  in  the  table  represent  1,  2,  3,  since  ^  =  yV>  and 
so  on.  This  term,  as  is  well  known,  is  always  negative,  and,  suppos- 
ing the  interpolations  performed,  we  have  the  data  here  given  with 
the  differences. 

17.  The  following  is  all  the  data  required,  except  constants,  to 
compute  a  solar  eclipse.  It  is  exactly  as  I  have  it  in  my  computing 
sheets  for  the  Nautical  Almanac,  and  the  reader  will  notice  that  I  do 
not  repeat  figures  unnecessarily.  The  reduction  from  time  to  arc  can 
be  performed  by  means  of  Table  I. 


17 


DATA   AND   ELEMENTS. 


35 


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36  THEORY  OF  ECLIPSES.  18 

18.  There  is  a  principle  in  the  theory  of  differences  that  I  have 
never  seen  in  any  of  the  ordinary  works  on  interpolation,  and  that 
is,  the  symbol  J  is  distributive.     We   have  for  multiplication  in 
algebra  m(a  ±b)  =  ma  ±  mb,  where  m  is  distributive.     Likewise  in 

differences, 

J(a±&)=  Ja±  J&.  (10) 

Now,  if  a  and  b  are  similar  terms  of  two  series  and  the  series  be 
added  term  by  term,  the  corresponding  difference  of  the  sum  will  be 
the  sum  of  those  differences.  This  is  illustrated  by  the  first  differ- 
ences of  the  two  right  ascensions  given  above ;  compare  their  sum 
term  by  term  with  the  differences  of  a  —  a' .  Likewise  compare  the 
second  and  the  third  differences.  We  can  check  a  number  of  addi- 
tions in  this*  manner.  This  principle  I  have  found  of  very  great  use 
in  certain  methods  of  computation  for  shortening  the  work. 

19.  There  is  also  another  point  similar  to  the  above,  the  moon's 
right  ascension  differences  very  irregularly  when  reduced  to  arc: 
this  depends  upon  the  fact  that  in  differentiating,  a  constant  factor 
in  the  primitive  remains  in  all  the  differences.     Thus, 

u  =  axn  '—  =  anxn~l  —  =  an(n  —  l)#n~2. 

dx  dx2 

Now  reduction  to  arc  is  simply  a  multiplication  by  15,  and  the  differ- 
ences of  a  are  seen  to  be  exactly  15  times  those  of  Q)  App.  R.  A. 
The  moon's  declination,  given  to  tenths  of  a  second,  causes  d  —  8'  to 
vary  irregularly  by  multiples  of  10,  which  are  slightly  affected  by 
the  sun's  differences.  Both  of  these  points  will  be  found  of  much 
assistance  on  a  subsequent  page. 

20.  Eclipse  Constants. — The  principal  of  these  are  TTO,  the  sun's 
parallax,  H,  the  sun's  mean  semidiameter,  and  k,  the  ratio  of  the 
moon's  equatorial  radius  to  that  of  the  earth.      They  have  been 
changed    from   time  to  time   as   more   accurate  values  have  been 
obtained. 

As  to  the  formulae  and  constants  used  in  past  years,  I  am  told  by 
Mr.  HILL,  that  Mr.  CHATJNCEY  WRIGHT,  who  first  computed  the 
Almanac,  used  his  own  formulae,  as  did  Mr.  HILL,  who  succeeded 
him  for  the  year  1874. 

CHAUVENET,  for  his  eclipses,  has,  through  an  oversight,  used 
both  values  of  ENCKE'S  Parallax  of  the  Sun.  For  the  Constants  of 
the  Cones,  on  page  448,  he  has  used  TTO  =  8".57116,  which  is  ENCKE'S 
second  value,  and  which  is  satisfactory;  but  on  page  452  he  gives  log 


20 


DATA   AND   ELEMENTS. 


37 


sin  TTO  —  5.61894,  which  belongs  to  ENCKE'S./^  value,  8.5776.  This 
latter  is  used  for  finding  6,  and  also  enters  into  the  constants  for  the 
cones ;  but  the  error  is  only  about  eight  (8)  units  in  seven-place  loga- 
rithms, and  hardly  affects  the  radii  of  shadows. 

Since  the  author  has  computed  the  eclipses,  the  following  constants 
have  been  used  for  the  eclipses  of  the  years  given : 


1883,  CHAUVENET'S  values  as  follows : 

TTO,  ENCKE'S  2d,  for  constants  for  cones,  8r/.57116 

1st,  for  6,  8   .5776 

J5T,  BESSEL'S  value,  959   .788 

k,   BURCKHAKDT'S  value,  0  .27227 

Kesulting  constants  for  cones  are 


log  sin  5.61861 
log  sin  5.61894 
log       2.9821753 
log       9.4349998 
Penumbra  log  C  7.6688033 
Umbra       log  G  7.6666913 


1896,  TTO, 
Constants  for  cones, 

1901,  when  NEWCOMB'S 
H, 

k 

Kesulting  constants, 

log 
new  tables 

log 

8.80 
C      7.6688314 

were  generally 
959.63 
0.272274 
C      7.6687600 

log 
log 

first 
log 
log 
log 

5.63006 
G  7.6666631 

used, 
2.9821038 
9.4350062 
G  7.6665914 

For  1902-3-4  only, 
k, 

Resulting  constants, 

For  1905  to  the  present  time, 
k,  the  previous  value  restored, 
Resulting  constants, 


0.272506    log       9.4353760 
log  C      7.6687609  log  G  7.6665909 

0.272274    log       9.4350062 
log  C      7.6687600  log  G  7.6665914 


1883-1900, 

1901, 

1902, 

1903-04, 
1905, 


Constants  of  Irradiation. 

Sun,  2".20         Moon,  2".50 
1   .15  2  .50 

1   .15  1   .50 


variable. 


1 

1   .50 


variable.    1   .50  +  0.000232  ?r 


The  two  latter  are  variable ;  their  values  may  be  taken  from  the 
small  tables  given  in  the  next  article.  These  quantities  are  not, 
strictly  speaking,  constants,  for  they  vary  with  the  state  of  the 
atmosphere,  the  telescope,  and  the  eye  of  the  observer,  and  the 
above  values  are  arbitrarily  assumed. 


38  THEORY   OF   ECLIPSES.  20 

The  following  constants  are  also  used  in  eclipse  computations : 

Eccentricity  of  the  terrestrial  spheroid  e, 

CHAUVENET  (BESSEL'S)  value,  0.0816967     log  8.9122052 
For  1882  and  following  years  CLARKE,  8.9152515 

log  1/1  —  e2  9.99853 

log       1  0.00147 

VI  —  e2 

These  constants  should  be  written  on  the  lower  edge  of  a  slip  of 
paper  for  future  use,  as  they  are  not  repeated  where  used  in  the 
examples;  and  placing  the  constant  above  any  quantity  in  the 
examples,  it  can  be  added  or  subtracted  without  rewriting  it. 

In  1892,  for  the  eclipses  of  1896,  Professor  NEWCOMB  substi- 
tuted the  parallax  8.80  for  ENCKE'S,  which  was  afterward  adopted 
by  the  Paris  Conference  in  1896  ;  and  it  was  first  used  in  the  body 
of  the  Almanac  for  1900.  The  changes  in  JThave  been  small,  and 
for  k  substantially  the  same  value  until  1899,  for  the  eclipses  of 
1902,  after  Professor  NEWCOMB'S  retirement,  when  the  value  was 
unfortunately  much  increased  to  0.272506  (log  9.4352760),  a  value 
derived  chiefly  from  occultations.  Compare  CHAUVENET,  I.,  551, 
note:  "According  to  OUDEMANS  (Astron.  Nach.,  vol.  li.,  p.  30),  we 
should  use  for  occultations  £  =  0.27264,  or  log  k  =  9.435590,  which 
amounts  to  taking  the  moon's  apparent  semidiameter  about  1".25 
greater  in  occultations  than  in  solar  eclipses."  Soon  after  the  eclipse 
of  1900,  May  28,  a  letter  from  the  English  Nautical  Almanac  office 
was  received  in  our  office,  stating  that  our  values  of  the  Besselian 
elements  gave  the  times  almost  exactly ;  whereupon  the  director 
promptly  restored  the  old  value  of  k  =  0.272274.  This  comparison 
with  our  Almanac  caused  the  English  Nautical  Almanac  to  adopt  a 
smaller  value  of  &,  very  nearly  equal  to  our  value. 

Wishing  to  know  for  the  present  work  the  exact  values  the  Eng- 
lish Nautical  Almanac  would  adopt,  Mr.  A.  M.  W.  DOWNING,  super- 
intendent of  the  English  Nautical  Almanac  office,  in  reply  to  my 
queries,  very  kindly  wrote  as  follows,  under  date  of  15  November, 
1901,  and  in  advance  of  the  publication  of  the  new  values  in  the 
Almanac  for  1905 : 

"  In  reply  to  your  letter  of  5th  instant,  the  constant  of  lunar 
parallax  used  in  the  eclipse  Computations  is  HANSEN'S,  viz.,  57' 
2".28.  The  value  of  log  k  is  given  on  page  629  (N.  A.,  1904),  and 
is  9.43542,  corresponding  to  mean  semidiameter  of  (D  15'  32.65. 

"  But  this  semidiameter  appears  from  the  observations  of  the  late 
total  eclipse  of  the  sun  to  be  too  large  for  use  in  eclipse  calculations, 


20  DATA    AND   ELEMENTS.  39 

and  in  1905  Nautical  Almanac  we  adopt  15'  31  ".47,  deduced  by  Dr. 
J.  PETEKS,  of  Berlin,  from  a  discussion  of  recent  observations  of 
eclipses  of  the  sun.  We  still  use  15'  32.65  for  occultations  of  stars 
by  the  moon." 

The  above  values  differ  slightly  from  ours,  but  the  two  Almanacs 
are  generally  now  in  accord  upon  this  point.  The  value  of  k,  giving 
the  above  semidiameters  for  eclipses  from  the  above  parallax,  is  k  = 
0.272178,  log  k  =  9.4348533. 

21.  Elements.  —  These  are  fundamental  values  for  an  eclipse, 
because  from  them  alone  the  times  can  be  computed  for  any  place 
by  the  method  of  semidiameters  ;  or  the  eclipse  plotted  graphically 
and  the  times  measured  by  scale.  These  elements  are  the  time  of 
conjunction  in  right  ascension,  and  for  this  time  —  the  right  ascensions, 
declinations,  hourly  motions,  semidiameters,  and  parallaxes. 

For  the  time,  which  is  when  («  —  a')  =  0,  we  have 


'- 


For  this  use  all  the  decimals  and  six  or  seven  places  of  logarithms. 
4j  is  here  a  mean  of  the  two  values  adjacent  to  Jw  and  the  term 

•  —  -  —  is  an  approximate  value  of  t,  which  can  be  gotten  mentally. 

(a  —  «r)  is  the  value  just  before  conjunction.     From  the  Data,  page 
35,  t  is  roughly  f|  =  0.8. 

+  J,  +34'    17//.85      —(«  —  a')    +  28'20".19  log   3.2304974 

-44  Kl-35)      +  0   .675 
—0.8  x  0.675     —       .540 

34'    17".985  log   3.3134423 

0.826143*  =  49W.5686  log  1  9.9170551 

.  •  .  V  8*  49™  34".12 

The  result  gives  decimals  of  an  hour,  since  the  data  are  given  by 
hours. 

For  this  time  the  right  ascensions  and  declinations  are  to  be 
interpolated  (Formula  9),  using  five-place  logarithms  and  second 
differences  for  the  moon,  which  can  be  taken  from  Table  III.  For 
the  above  decimal  of  an  hour  we  find  the  coefficient  for  J2  —  0.072, 
The  correction  for  J2  can  be  gotten  mentally.  The  right  ascensions 
must  agree  exactly,  which  will  check  the  time,  at  least  the  logarithm. 
The  declinations  can  be  checked,  if  desired,  by  interpolating  d  —  d'9 
which  should  equal  their  difference  very  closely. 


40 


THEORY   OF   ECLIPSES. 


21 


Next,  interpolate  log  r'  for  the  time  t ;  it  can  generally  be  done 
mentally.     Then  the  true  semidiameter  and  parallax  are 


H 

rf 
KO 
r' 


(12) 
(13) 


Constant  of  Irradiation  for 

Sun's  Semidiameter. 

logr'. 

Irradiation. 

0.0094       94 

1".83 

70      f, 
46      l\ 

1    .84 

1    .85 

23      ;* 

1    .86 

oo    r( 

1    .87 

9.9976      «J 

1    .88 

53      g 

1    .89 

30      09 
9.9908" 

1    .90 
1    .91 

These  can  be  checked  by  the  values  from  the  Nautical  Almanac, 

interpolating  for  the  time  t,  and  deduct- 
ing from  the  semidiameter  the  constant 
of  irradiation  mentioned  above.  This 
variable  value  for  1903  to  the  present 
time  can  be  taken  from  annexed  table. 

Next,  interpolate  the  moon's  parallax 
to  three  decimals  for  the  time  t  by  natu- 
ral numbers,  and  get  the  logarithms  in 
seconds  ;  then  the  true  semidiameter, 

8=fa.  (14) 

Check  this  by  interpolating  the  moon's 
semidiameter  from  the  Nautical  Almanac, 

using  second  differences  for  an  interval  of  12  hours  (Table  III.),  and 
deducting  the  constant  of  irradiation  from  the  annexed  table,  or 
according  to  the  value  of  the  constant  used.  This  should  agree 
within  about  ".05,  because  the  Almanac  gives  only  to  tenths  of  a 
second,  which  may  be  ".05  in  error.  The  value  here  computed  will 

be  more  correct  than  that  in  the  Alma- 
nac, if  no  mistake  is  made. 

The  hourly  motions  are  gotten  as  fol- 
lows :  A  formula  can  be  used,  but  is 
unnecessary.  For  the  moon  in  declina- 
tion, we  want  the  first  difference  at  the 
time  t,  which  is  between  8  and  9\  In 
the  table  of  data  —11'  31".6  is  the 
motion  at  8.30,  and  —  11'  33".6  the 
motion  at  9.30.  cf  is  at  Sh  50m  nearly ; 
that  is,  20m  after  the  half  hour;  the 
change  during  the  interval  between  the 
half  hour  is  J2  —  2".0 ;  in  60™,  t  is  f# 
of  this  interval;  and  J  of  -20,  or 

-0.7,  being  added  algebraically  to  —11'  31".6,  we  have  —11' 
32".3  for  the  hourly  motion  required. 


Constant  of  Irradiation 

for 

Moon's  Semidiameter. 

Constant 

it. 

Irradiation. 

52' 

50" 

—  0".73 

—  V 

.50 

53 

0 

0 

.74 

1 

.50 

54 

0 

0 

.75 

1 

.50 

55 

0 

0 

.77 

1 

.50 

56 

0 

0 

.78 

1 

.50 

57 

0 

0 

.79 

1 

.50 

58 

0 

0 

.81 

1 

.50 

59 

0 

0 

.82 

1 

.50 

60 

0 

0 

.83 

1 

.50 

61 

0 

0 

.85 

1 

.50 

62 

0 

—  0 

.86 

—  1 

.50 

21  ECLIPSE  TABLES.  41 

Hence  we  have  the  elements — 

Greenwich  mean  time  of  cf  in  R.  A.,  1904,  Sept.  9*  8*  49m  34M 

Sun's  E.  A.                   11*  II"1  5*.39  Hourly  motion,                  9.00 

Moon's  R.  A.                11  11    5 .39  "                           2  26".16 

Sun's  dec.                +  5°  14'  56".29  "           "              -  56".7 

Moon's  dec.             +5      4  30  .69  "           "        —  IV  32".3 

Sun's  equa.  hor.  parallax,         8'.74  True  semidiameter  15'  53".17 

Moon's       "             "         61  22'.957  "              "            16  43  .63 

I  may  add  here  that  the  time  of  new  moon  is  not  the  same  as  that 
we  have  computed  above,  being  the  time  of  conjunction  with  the  sun 
in  longitude. 


SECTION    IV. 

ECLIPSE  TABLES. 

22.  WITH  this  section  begins  the  main  computation  for  the  Eclipse 
Tables,  with  which  all  the  rest  of  the  calculation  is  made.     Seven- 
place  logarithms  are  to  be  used  until  I  and  log  i  are  gotten  ;  then 
five-place  for  the  rest  of  the  work. 

The  computer  should  prepare  in  advance  the  general  constants  for 
all  eclipses  on  the  lower  edge  of  a  slip  of  paper,  so  that  it  may  be 
placed  above  other  quantities  it  is  to  be  combined  with  ;  and  on  another 
slip  certain  constants  which  change  ivith  each  eclipse.  Constants  are 
generally  not  given  in  the  examples  following.  The  reader,  however, 
following  this  suggestion,  by  the  aid  of  the  formulae  will  doubtless 
have  no  difficulty  with  the  examples,  which  are  here  given  just  as 
they  stand  on  the  author's  computing  sheets. 

23.  We  recapitulate  the  general  constants  used  in  the  examples 
as  follows  : 

log  TTO,  8.80  0.9445 

log  sin  TTO,  8.80  5.63006 

logJT,  959.63  2.9821038 

log  &  *  0.272506  9.4353760 

(Penumbra,  7.6687609 
Constants  for  cones  {  Umbra)        7.6665909 

9.99853 
0.00147 


*  As  stated  above,  this  value  was  used  only  for  the  years  1902-3-4.     The  old  and 
more  correct  value  0.272274  has  now  been  restored. 


42  THEORY   OF   ECLIPSES.  23 

A  few  other  constants  used  only  in  particular  places  will  be  given 
in  the  text  where  they  occur,  and  they  can  be  added  to  the  slip  of 
paper  on  which  the  above  should  be  written,  as  they  are  also  used 
for  all  eclipses. 

24.  I  propose  to  give  first  the  formulae  in  groups  of  related  quan- 
tities, commenting  upon  them  and  their  use  in  connection  with  the 
example  which  follows.    In  this  order  the  computation  will  be  given 
in  an  unbroken  order.    And  following  this  will  be  given  the  Eclipse 
Tables  which  result  ;  these  are  also  necessarily  explained  with  the 
formulae.     The  small  terms  of  a  and  d,  and  the  quantities  used  only 
for  them,  may  be  computed  with  five-place  logarithms. 

25.  The  Main  Computation  and  CHAUVENET'S  Formulae.  —  In  this 
formula,  G  is  the  distance  between  the  centres  of  the  sun  and  moon, 


And  two  auxiliaries  are  assumed,  such  that 

G  ,    r 

—  =  a        and  —  =6. 

r'  r' 

If  we  deduct  the  third  of  these  equations  from  unity  in  each  number, 
we  have 


And  since  we  have  generally  sin  nr  =  —  ~>  which  is  a  more  correct 
formula  than  No.  (13),  we  have 

.  _   jr  __  sin  icr        sin  TTO 

rf       sin  TT        r'  sin  TT 

For  the  eclipse  hours  of  Greenwich  mean  time,  which  were  deter- 
mined above,  commence  the  computation  with  the  following  formulae  : 


r  sn  TT 
0=1-6  (16) 

a  =  a>  --  —  cos  d  sec  *'  (a  —  a')  (17) 

1  —  b 

(d  —  d')  (18) 


1  —  b 


25  ECLIPSE   TABLES  43 

a  and  d  are  the  right  ascension  and  declination  of  the  point  Z  of 
the  celestial  sphere. 

The  quantity  1  —  b  is  not  required  separately,  and  by  Table  V. 

the  quantity  — '—-  may  be  taken  out  at  once  with  the  argument  ->  or 
the  compliment  of  log  6. 

26.  The  second  terms  of  formulae  (17)  and  (18)  are  very  small — 
not  over  10"  or  15" — and  the  numbers  found  from  the  logarithms 
are  transferred  to  the  Eclipse  Tables  and  the  sign  changed.     Then  a 
and  b  are  gotten  there  and  differenced.     As  soon  as  transferred  to 
the  Eclipse  Tables  all  quantities  should  be  differenced,  and  the  last 
difference  should  run  at  least  as  smoothly  as  in  the  present  example. 

(a  —  a)  and  (d  —  d)  are  next  found  and  differenced,  then  J(a  —  d) 
and  (d  -j-  d).  The  reader  will  remark  that  the  difference  of  the  small 
terms  is  equal  to  the  third  small  term ;  thus, 

(a  —  ar)  —  (a  —  a')  =  a  —  a  ; 

and  similarly  for  d  —  d.    This  comparison  will  check  the  intermediate 
work.  «. 

27.  The  Greenwich  hour  angle  of  the  principal  meridian  passing 
through  the  point  Z  at  any  time  is  given  by  the  equation 

f,,  =  B  +  h'  -  a.  (19) 

This  is  done  on  the  computing  sheets.  h'  is  the  sidereal  equiva- 
lent of  the  mean  time  eclipse  hours  h.  CHAUVENET  adds  8  -f  h' 
in  time  and  then  reduces  to  arc.  The  better  way  is  to  reduce  the 
sidereal  time  to  arc,  which  was  done  on  the  table  of  data  already 
given  ;  then  use  Table  VI.,  which  gives  the  sidereal  equivalent  in 
arc.  The  addition  is  made  on  the  computing  sheets,  then  fa  is  given 
in  the  next  column. 

The  hourly  variation  of  fa  or  the  change  in  1  hour  in  seconds  of 
arc  and  in  parts  of  radius, — 

log  thf  =  log  ( 4"!  sin  1")  (20) 

can  be  computed  or  taken  at  once  from  Table  VII.  with  the  argument 
A  fa.   It  is  constant  for  one  eclipse,  and  is  needed  only  to  five  decimals. 
For  the  Nautical  Almanac  change  of  fa  for  1  minute  of  time  in  min- 
utes of  arc  is 

log  A  p.  now  called  log  /*'  =  log  -^—  (21) 

3600 


44  THEORY   OF   ECLIPSES.  27 

A fa  is  here  given  in  seconds,  and  this  is  the  change  of  [JLV  in  minutes 
of  arc  for  one  minute  of  time.  Given  to  four  decimals  only,  and  it 
can  be  taken  from  Table  VII.  at  the  bottom. 

log  /  =  log  O/  cos  d)  (22) 

Required  to  four  decimals  only. 

This  latter  is  from  CHAUVENET'S  formula  512,  the  rest  of  which 

is  not  needed  at  all, 

/  sin  F  =  d' 

f  cos  F  =  /*/  cos  d 
in  which  placing  cos  F=  unity,  we  have  the  above  form. 

28.  Proceeding  with  the  computation, 

r=^-  (23) 

sin  T: 

x  =  r  cos  d  sin  (a  —  a)  (24) 

y  =  r  sin  (d  —  d)  cos2*  (a  —  a)  +r  sin  (d  +  «f)  sin2*  (a  —  a)     (25) 

z  =  r  cos  (d  —  d)  cos2*  (d  —  a)  -  r  cos  (d  +  d)  sin'J  (*  -a)      (26) 

These  are  the  coordinates  of  the  axis  of  the  cone  of  shadow,  and 

they  will  be  found  rather  difficult  to  get  correctly.     A  partial  check 

upon  them  is  given  under  the  article  At  Noon,  but  it  is  not  of  much 

value  here,  and  given  there  chiefly  as  it  leads  up  to  other  matters. 

The  second  terms  are  small  and  computed  with  five-place  loga- 
rithms ;  but  added  to  the  first  terms  by  ZECH'S  Addition  and  Sub- 
traction  Logarithms. 

In  the  example  following,  and  wherever  else  such  tables  are  used, 
the  expression  I — /  means  the  difference  of  the  two  logarithms,  usu- 
ally expressed  in  the  tables  as  "log  —  log-"  In  addition  this  is 
called  A,  and  the  table  gives  B.  In  subtraction  it  is  called  B,  and 
the  table  gives  A. 

Log  r  is  given  here,  but  is  not  on  my  computing  sheets,  but  on  a 
second  slip  of  paper  containing  these  quantities,  which  are  constant 
for  one  eclipse,  but  change  slowly  with  the  season. 

29.  Copy  log  x  in  the  Eclipse  Tables  and  difference  it,  pass  to 
numbers  to  six   decimals  and   difference  them  also.     Proceed  the 
same  with  log  y — but  log  z  is  not  needed  among  the  tables,  so  that 
it  is  not  copied.     It  is  required  below  in  this  computation,  and  it 
should  difference  smoothly  if  TT  is  interpolated  to  three   decimals. 

x  is  seen  to  difference  rather  irregularly  in  the  third  differences  > 
tracing  this  back,  we  find  it  in  (a  —  a)  and  still  further  back  in  a  in 
arc ;  which,  we  said  in  the  previous  section,  varies  by  multiples  of  1 5, 
caused  by  the  reduction  from  time  to  arc,  To  ascertain  whether 


29 


ECLIPSE  TABLES. 


45 


Assuming 

various  values  of  r  and  d. 


Differential  Formula  for  x. 

Logr 

0° 

10° 

20° 

30° 

1.75 

2.73 

2.68 

2.56 

2.36 

1.76 

2.79 

2.75 

2.62 

2.42 

1.77 

2.86 

2.81 

2.68 

2.47 

1.78 

2.92 

2.88 

2.74 

2.53 

1.79 

2.99 

2.94 

2.80 

2.59 

1.80 

3.06 

3.01 

2.87 

2.69 

1.81 

3.13 

3.08 

2.93 

2.71 

the  irregularities  of  x  are  due  wholly  to  this  cause  or  to  small  errors 
from  other  sources  I  have  devised  the  following  differential  formula 
from  No.  24.  Differentiating,  with  x  and  a  variable,  we  have 

dx  =  —  r  cos  d  cos  (a  —a)  da  x  sin  V  (27) 

a  =  0".01,  this  table  has  been  constructed  from  the 
And  to  use  it  the  reader  is  referred 
to  the  remarks  following  the  Data  of 
this  eclipse  given  in  Art.  18,  Eq.  10. 
For  this  eclipse  log  r  =  1.748,  d  =  5°, 
whence  from  the  annexed  table  we 
have  2.69  ;  J3(«  —  a)  in  the  tables 
varies  thus,  —  13  --2  +1  +14. 
If  we  add  to  it  the  same  series  with 
contrary  signs,  +13+2  —  1  —  14, 
it  will  all  reduce  to  0.  Now  every 
0".01  affects  x  by  2.69  units  in  the 
sixth  place  of  decimals.  Multiply  this 
latter  series  by  this  constant  and  add  it  to  Azx ;  the  sum  should 
reduce  to  zero,  or  more  likely  become  some  constant ;  we  thus  have 

J3z     _  24     —  54     —  56     —  96 
-2.694  (a  —  a)     —35     —    5     +3     +38 

Sum     —  59     —  59     —  53     —  58,   a  constant,  nearly, 

which  shows  that  the  irregularity  of  x  depends  wholly  upon  omis- 
sion of  a  third  decimal  in  a  in  time. 

A  formula  for  y  is  similarly  derived  and  here 
tabulated,  and  a  formula  for  z  may  also  be  de- 
rived ;  but  this  latter  is  of  little  use,  since  z  differ- 
ences quite  smoothly,  or  should  do  so  if  TT  is  inter- 
polated to  three  decimals. 

30.  Epoch  Hour. — This  must  now  be  assumed. 
It  may  be  taken  at  any  of  the  hours,  but  should 
be  near  the  middle  of  the  eclipse,  which  is  not 
yet  known,  so  we  may  take  it  near  the  time  of 
conjunction.  Although  the  Epoch  Hour  may  be 
assumed  anywhere,  an  integral  hour  must  be 
taken,  on  account  of  the  method  here  given  for 
computing  the  hourly  motions ;  and  when  once 
assumed,  it  cannot  be  changed  without  recom- 
puting the  mean  hourly  motions.  The  9th  hour  is 
assumed  in  the  present  eclipse,  and  for  this  time 
the  hourly  motions  of  x  and  y  are  to  be  gotten  in  the  Eclipse  Tables. 


Differential  Formula 

for  y. 

Log  r'. 

Equation. 

1.750 

2.73 

1.755 

2.76 

3 

1.760 

2.79 

3 

1.765 

2.82 

3 

1.770 

2.85 

3 

1.775 

2.88 

3 

1.780 

2.92 

4 

1.785 

2.95 

3 

1.790 

2.99 

4 

1.795 

3.02 

3 

1.800 

3.06 

4 

1.805 

3.09 

3 

1.810 

3.13 

4 

46  THEOKY  OF   ECLIPSES.  30 

The  Epoch  Hour  is  the  standard  from  which  all  the  computed  times 
depend.  We  must,  therefore,  know  the  changes  of  x  and  y,  counted 
from  this  hour.  For  example,  the  change  between  9  and  10  hours 
is  the  average  rate,  which  would  be  the  change  at  9.30.  Likewise 
the  change  from  9A  to  11A  would  be  the  mean  or  average  change 
which  is  the  mean  of  the  two  differences  at  9.30  and  10.30.  Also 
the  change  from  9*  to  12*  is  one-third  of  the  three  differences.  Thus, 
between  9fc  and  any  other  time  these  are  seen  to  be  the  average  or 
mean  motions,  and  hence  properly  called  by  CHAUVENET,  p.  451, 
line  3,  the  Mean  Hourly  Changes.  It  would  be  well  to  take  for 
these  the  notation  xQf  yQf  .  They  are  used  by  the  computer  of  the 
eclipse  throughout,  and  are  the  quantities  given  by  CHAUVENET  on 
page  455. 

The  formula  in  this  case  is  that  for  finding  a  first  difference  for 
any  given  argument  (CHAUVENET'S  Astronomy,  I.  90)  : 

f'T=  i  (4  -  i-4  +  A4  -  etc.). 

Here  w  is  the  fraction  of  the  interval,  which  is  unity  in  our  case  ; 
we  may  omit  the  third  term  ;  4  and  J3  are  to  be  taken  as  mean 
value  of  the  two  adjacent  differences;  this  half  sum  is  usually 
expressed  by  \2A.  Hence,  for  our  use  we  have  for  the  mean  hourly 
changes  — 

For  the  epoch  hour, 


(28) 
And  for  the  other  hours,  the  second  differences  not  being  used, 

At  6A  \  of  the  three  differences  below  it. 

7A|-     "       two 

8*  the  first 

10*       "  "         above  it. 

IP  |  of  the  two 
12*  t     "       three 

On  the  other  hand,  the  astronomer  who  wishes  to  observe  the 
eclipse  and  compute  the  times  beforehand  knows  nothing  of  our 
Epoch  Hour,  but  assumes  some  hour  near  the  time  when  the  eclipse 
begins  at  his  station,  say  7A,  and  another  observer  llfe,  according  to 
his  location,  and  they  want  to  know  the  actual  change  at  the  hours 
they  select.  These  we  may  term  the  absolute  hourly  changes,  which 
are  given  by  CHAUVENET  on  page  464,  and  are  also  the  quantities 
given  in  the  BESSELIAN  Elements  in  the  Nautical  Almanac,  after 


30  ECLIPSE   TABLES.  47 

being  divided  by  60  to  give  the  changes  in  one  minute.  This  is 
taking  w  =  60  in  CHATJVENET'S  formula  above  quoted. 

The  absolute  hourly  motions,  which  may  be  noted  at  x'  y',  are 
computed  for  each  hour  by  the  formula  (28)  j  and  for  this  the  differ- 
ences of  x  and  y  will  generally  have  to  be  extended  above  and  below, 
as  in  the  example.  The  numbers  should  all  here  be  carried  out  to 
six  decimals ;  though  the  last  is  not  very  accurate,  yet  it  had  better 
be  retained.  Get  the  logarithms  of  these  to  five  places ;  these  are 
omitted  from  the  printed  page  for  lack  of  room. 

31.  The  following  are  called  the  constants  for  cones  for  eclipses  : 

Penumbra  C=  sin  H  +  k  sin  TTO.  (29) 

Umbra       d  =  sin  H  —  k  sin  TTO.  (30) 

They  should  be  carefully  computed  with  seven  places  of  logarithms 
and  written  on  the  lower  edge  of  a  slip  of  paper  for  use  in  all 
eclipses.  They  are  changed  only  when  more  accurate  values  of 
the  constants  entering  are  obtained. 

C 

For  Penumbra,  sin  /  =  — >  (31) 

r'g 

•-•  +  -47  (32) 

sin/ 

t=tan/,  (33) 

I  =  ic.  (34) 

For  Umbra,       sin  /  =  -^->  (35) 

c^z-^-,  (36) 

sin/! 

ij  =  tan  /,  (37) 

^1=^.  (38) 

In  these  formulae  /  is  the  angle  of  the  cone  of  shadow,  I  and  ^ 
the  radii  of  the  Penumbra  and  shadow  on  the  fundamental  plane ; 

z,  the  distance  of  the  moon's   centre ;    >    the  distance  of  the 

Bin/ 

vertex  of  the  cone  of  shadow  from  the  moon ;  and  therefore  c,  the 
distance  of  the  vertex  of  the  cone  above  the  fundamental  plane. 
Hence,  we  have  the  species  of  the  eclipse  : 

When  the  sign  of  li  for  Umbra  is  positive,  the  eclipse  is  annular. 
"      I,          "          is  negative,          "       is  total. 

I  for  Penumbra  is  always  positive. 


48  THEORY   OF   ECLIPSES.  31 

Tan  /  can  readily  be  found  from  the  sine  by  the  annexed  table, 
which  gives  log  sec  i  for  the  corresponding  values  of  log  sine. 

If  addition  and  subtraction  logarithms  are  used,  log  —  log  for 
Umbra  is  found  at  once  from  that  of 
Penumbra,  and  is 


log  sin/. 

7.6603—  48 
7.6649  —  95 
7.6696  —  *  41 
7.6742—  88 


log  sec/. 

0.0000046 
47 
48 

0.0000049 


Greater  than  for  Penumbra  when  the 

eclipse  is  total, 
Less  than  for  Penumbra  when  the  eclipse 


is  annular, 

in  each  case  differing  by  the  difference  of  the  constants  for  cones. 
Both  formulae  should  be  computed,  using  this  as  a  check. 

Tabulate  log  I  and  log  ^  to  six  places  of  logarithms  and  pass  to 
numbers,  preferably  six  places.  Tabulate  log  i  log  i\  to  five  places 
of  logarithms,  and  for  these  latter  the  natural  numbers  are  not 
needed.  Difference  them  all. 

32.  Compute  the  remainder  of  the  tables  with  five-place  loga- 
rithms and  to  seconds  of  arc. 

pl  sin  dl  =  sin  d,    )  ^ 

/>!  cos  dv  =  cos  d  1/1  — e2,  / 

These  quantities  are  used  to  take  account  of  the  compression  of 
the  earth  instead  of  using  d  and  p. 

In  equations  of  this  form,  in  the  first  member  the  first  factor  is  a 
linear  quantity,  and  ALWAYS  positive ;  the  second  factor  is  an  angle, 
of  which  the  signs  are  determined  from  the  second  member,  and 
hence  the  quadrant.  In  the  present  case  dl  differs  but  little  from 
the  declination  of  the  sun.  Tabulate  these  quantities  and  difference 
them  ;  log  cos  d1  should  also  be  tabulated,  since  it  is  frequently  used 
in  the  succeeding  calculations.  Log  sin  d1  we  will  pass  over  for  the 
present. 

V  =  —  yf  -f  ^x  sin  d  (40) 

Penumbra    cr  =  x'  -f  n'-y  sin  d  +  &'$  cos  d  (41) 

Umbra        c\  =  xf  +  n'y  sin  d.  (42) 

The  Absolute  Hourly  Motions  are  to  be  used  here  for  the  first 
terms.  The  only  difference  between  cf  for  Umbra  and  Penumbra  is 
in  the  third  term,  which  for  Umbra,  on  account  of  the  small  value 
of  I,  is  almost  or  quite  insignificant,  and  is  omitted.  This  term  for 
Penumbra  is  nearly  constant  for  one  eclipse,  and  varies  from  about 
0.000610  to  0.000680,  and  this  affects  the  angle  E  (next  to  be  com- 


32  ECLIPSE  TABLES.  49 

puted)  some  10'  or  15'  of  arc;  and  yet  CHAUVENET  computes  E 
for  Penumbra  only  and  uses  that  for  the  limits  of  Total  Eclipse.  It 
is,  however,  in  a  formula  where  it  has  little  effect.  But  the  better 
way  is  to  compute  cr  and  E  for  the  Umbra,  and  use  them  for  the 
Penumbra,  where  less  accuracy  is  necessary,  so  that  equation  (41) 
above  may  be  wholly  neglected,  and  (42)  substituted  in  its  place. 

Compute  the  second  terms  and  add  by  natural  numbers.  Trans- 
fer the  numbers  of  bf  to  the  tables  and  also  get  the  logarithm  there. 
cf  is  not  needed  among  the  tables,  but  may  be  placed  there  and 
differenced,  or  differenced  on  the  line  below  in  the  computation. 

The  hourly  motions  of  the  above  quantities  are  simply  the  loga- 
rithms of  the  mean  of  adjacent  differences  at  the  epoch  hour,  one 
value  of  each  given  to  four  decimals  of  logarithms. 

60"=iZ4&'.  (43) 

c0"  =  \ZA&'.  (44) 


The  quantities  bf  and  c'  are  the  relative  motions  of  the  surface  of 
the  earth  and  the  shadow ;  and  the  angle  E  next  to  be  computed  is 
the  angle  of  this  resultant  path, 

e  sin  E  =  i 
c  cos  E  = 

E  may  be  tabulated  to  the  tenth  of  a  minute  and  differenced. 
And  in  equations  of  this  form  that  are  important  I  generally  com- 
pute the  linear  quantity  by  both  equations.  In  these  equations  I 

get  first  log  6,  and  then  log  -  >  and  tabulate  the  latter,   because  e  is 

e 

used  only  in  the  denominator.     The  natural  numbers  of  e  are  not 
required. 

33.  Finally,  to  take  account  of  the  compression  of  the  earth  : 

.»-£        .  ^ 

*-*•  W 

The  former  of  these  can  be  found  when  required,  but  the  latter 
should  be  computed  from  the  Mean  Hourly  Changes  of  y  in  the  tables. 

34.  The  following  formulae  are  given  in  CHAUVENET  ;  but  are 
not  generally  required,  and  need  not  be  computed. 


50  THEORY   OF   ECLIPSES.  34 

Classed  with  bf  and  c'  we  have  for  Penumbra  and  Umbra  : 

a'  =  —  V  —  n'ix  cos  d  (48) 

Classed  with  p.', 

d'  =  ||sml"  (49) 

Classed  with  E  and  e, 

fSmF=d'  \ 

/COsF-Jl'COSCfJ 

Classed  with  d  and  log  pl  , 

Pi  sin  da  =  sin 


^ 

/02  COS     2  =  COS 

35.  The  quantities  in  the  tables  which  vary  rapidly  must  now  be 
interpolated  to  every  10  minutes.  The  interpolation  is  generally 
simple,  and  need  not  be  done  here  in  this  example.  The  quantities 
are  as  follows  : 

x,  y,  I,  I,  each  to  five  places,  natural  numbers. 

Hi  and  dl  to  seconds. 

Log  sin  dl  from  the  computation  to  five  places  of  logarithms. 

Log  bf  to  four  places  of  logarithms  if  possible.    If  not,  then 

the  natural  numbers  to  five  places. 
E  to  tenths  of  a  minute. 

Log  -  to  five  places  of  decimals. 

6 

In  a  computation  like  the  present,  where  signs  change  frequently, 
it  is  a  good  plan  to  write  the  -f  sign,  as  well  as  the  —  ,  before  each 
quantity  at  the  head  of  a  line  or  column.  But  in  the  middle  of  a 
line  or  column  the  sign  may  be  omitted,  except  where  they  change, 
and  also  the  characteristics  and  other  figures  which  do  not  change. 
In  using  addition  and  subtraction  logarithms,  I  —  Us  the  numerical 
difference  of  the  logarithms  ;  which  is  called  A  in  addition,  and  B 
in  subtraction  ;  and  the  other  letter,  B  or  A,  is  then  taken  out  from 
the  table. 

KEMARK.  —  The  author  wishes  it  understood  throughout  this  work  that  constants 
and  other  quantities  which  are  repeated  in  the  formulae  are  not  given  or  repeated  in  the 
examples.  They  should  be  written  on  a  slip  of  paper,  and  used  when  required,  as 
suggested  in  Article  20.  There  are  two  species  of  constants  —  those  which  are  abso- 
lutely constant,  and  those  which  are  constant  for  one  eclipse. 


36 


ECLIPSE  TABLES. 


51 


36.  THE  MAIN  COMPUTATION  FOB  TABLES. 

TOTAL  ECLIPSE, 

1904,  SEPTEMBER  9. 

Formulae. 

7*.                        To  =  9».                       12». 

(15)                                sin  TT 

+  8.2517542           8.2517475      +  8.2517063 

r'  sin  TT 

+  8.25470              8.25468          +  8.25462 

log  6 

-f  7.37536               7.37538          +  7.37542 

l-l,B 

2.62464               2.62462              2.62456 

(16)                                  ji  =  log(l:l—  6) 

4-  0.0010320           0.0010320      +  0.0010321 

(17)                                 cos  d 

4  9.9980500             .9983165      4-  9.9986816 

sect' 

4  0.00185                .00182          4-  0.00179 

log  (a  —  a') 

—  3.57512          4-  2.55348          +  3.81445 

—  (a  —  a')  =  sum,  log 

—  0.95141          +9.93003          +1.19139 

(18)                                  log  6  —  6' 

4-  2.72552          —  2.86695          —  3.42363 

—  (d  —  6')  =  sum,  log 

4-  0.10191          —  0.24336          —  0.80010 

(23)                                 logr 

+  1.7482458      4-  1.7482525      +  1.7482937 

(24)  (Check  Eq.  27)     sin  (a  —  a) 

—  8.2616996      +7.2400759      48.5009816 

x  is  0  at  cf.           log  x 

—  0.0079954      +  8.9866449      +  0.2479569 

(25)                                 sin  (6  —  d) 

+  7.4121222      —  7.5535539      —  8.1102273 

cos2^*  —  a) 

+  9.9999638      +  9.9999996      +  9.9998909 

Log  r  is  given  above,     log  (1  ) 

+  9.1603318      —9.3018060      —9.8584119 

sin(rf  +  c?) 

+  9.26884          +  9.25189          +  9.22492 

sin2  £(a  —  a) 

+  5.92138          +  3.87806          +  6.40001 

log  (2) 

+  6.93847          +  4.87820          +  7.37322 

*-/, 

.42.22186          B  4.42361           £2.48519 

£orA       B  +  0.0025979  ^1  —  0.0010164  ^4  —  0.0014233 

logy 

+  9.1629297      —  9.3017896      —  9.8569886 

(26)  All  the  terms  of    cos  (6  —  d) 

+  9.9999986           9.9999972      +  9.9999639 

this    equation    are    log  (3) 

+  1.7482082             .7482493      +  ]  .7481485 

positive  always.         cos  (<5  +  d) 

+  9.99238                .99296          +  9.99379 

Log  r  in  this  formula  \,      /,\ 
is  given  above.         / 

+  7.66201              5.61927         +  8.14209 

1  —  /always  B  for  Z.     Z  —  /,.B 

4.08620              6.92898              3.60606 

u4 

0.0000356             .0000003           0.0001076 

Log  Z  is  always  -J-.  ~\ 

Natural    numbers  y  log  Z 

+  1.7481726            .7482490      +  1.7480409 

are  not  required,     j 

(31)                                 log(l:r'0) 

+  9.9980898            .9980992      +  9.9981135 

(Constant,  Eq.  29).        sin/ 

+  7.6668507             .6668601      +  7.6668744 

(32)                                 (*:rin/) 

+  1.7685253             .7685159      +  1.7685016 

J-M 

0.0203527             .0202669           0.0204607 

5 

0.2909729             .2910147           0.2909201 

logc 

+  2.0594982             .0595306      +  2.0594217 

(33)                                logi 

+  7.6668554             .6668648      +  7.6668791 

f  34)  For  Penumbra  J\Jo   j 
is  always  +.            I 

+  9.7263536             .7263954      +  9.7263008 

(35)  (Const't,  Eq.  30.)    sin/, 

+  7.6646807            .6646901      +  7.6647044 

(36)                                (*:  ain/0 

+  1.7706953            .7706859      +  1.7706716 

J  —  Z,  £ 

0.0225227             .0224369           0.0226307 

4 

1.2963765             .2979916           1.2943525 

logcj 

—  0.4743188             .4726943      —0.4763191 

(37)                               logi, 

+  7.6646854             .6646948      +  7.6647091 

(38)  Total  Eclipse.          log^ 

—  8.1390042        —  .1373891      —  8.1410282 

52 


THEORY  OF   ECLIPSES. 


37 


CO  CO  r-  1  *O  lO  OO  CO 
O5  rH  rH  CO  O5  O5  CO 
?  CO  O5  O5  O5  O5 
to  i—  (  <M 


«O  -H? 
CO  CO 


rH  CO  iO  <M  O5  O 
CO  ^f    iO  CM  O5 


O5  GO  O5  O  O 

I  1+  + 

CO  t*  CO  O5  O  rH  CO 

iT*1 


37 


ECLIPSE  TABLES. 


53 


CO  CO  O  OO  t>  1C 
CO  CO  1C  <M  TH  <N 


os  co  co  ic  cq  co 


54 


THEORY  OF   ECLIPSES. 


36 


(39) 


ART.  36.  Computation.     (Continued.) 
sin  d 
cos  d 


cosdi 
Log  pl  is  always  -{-.       log  pl 

(40)  //!  x  sin  d 

y'   is    found    in    the    Numbers 

Eclipse  Tables  b' 

(42)  «!  y  sin  d 

*'  is  found   in   the|N     ^ 

Eclipse  Tables.        / 
c'  is  always  -{-•  tf 


(45) 


log  V 
logc' 


E 


loge 
cosJ? 


4-  8.96368 

8.96116 

4-  8.95732 

4-  9.99815 

.99818 

4-  9.99821 

4-  9.99668 

.99671 

4-  9.99674 

4-  8.96700 

.96445 

8.96060 

4-5  17  43 

5  15  52 

4-5  13  5 

4-  8.96514 

8.96261 

8.95871 

4-  9.99854 

855 

9.99855 

4-  9.99814 

9.99816 

9.99820 

4-  9.99854 

9.99855 

4-  9.99854 

—  8.38983 

4-  7.36591 

4-  8.62340 

—  0.024538 

4-  .002322 

4-  0.042015 

4-  0.148335 

4-  .175311 

4-  0.215074 

4-7.54472 

7.68106 

—  8.23244 

4-  0.003505 

.004797 

—  0.017078 

4-  0.561244 

.552990 

4-  0.540338 

4-  9.17124 

.24381 

4-  9.33259 

4-  9.74915 

.74272 

4-  9.73266 

4-  9.42209 

.50109 

9.59993 

4-14  48  16 

4-  17  35  23 

4-  21  42  18 

4-  9.40743 

4-  9.48029 

4-  9.56800 

9.76381 

9.76352 

9.76459 

4-  9.98534 

4-  9.97920 

4-  9.96806 

4-  0.23619 

4-  0.23648 

-f  0.23540 

Log  e  is  always  4~- 

The  numbers  of  formulas  omitted  above  are  given  in  the  Tables,  in  which  further 
but  brief  computations  are  made. 

38.  Geometrical  Illustration  of  the  Foregoing  Quantities  and  the 
Eclipse  Generally. — The  Fundamental  Plane  in  the  theory  of  eclipses 
to  which  all  quantities  are  referred  we  will  take  as  the  plane  of 
the  paper.  It  is  the  plane  XY  of  the  coordinate  axes  which 
intersect  in  the  centre  of  the  earth.  The  radius  of  the  earth 
is  the  unit  of  measure,  and  has  been  the  unit  of  all  the  quantities 
and  decimals  computed  for  the  Eclipse  Tables. 

Fig.  2,  Plate  I.,  is  an  orthographic  projection  of  our  example,  the 
Total  Eclipse  of  1904,  September  9.  In  this,  as  well  as  in  the 
other  principal  figures  of  this  work,  the  unit  of  measure  is  assumed 
to  be  two  and  one-half  inches ;  a  scale  of  40  parts  will  then  give 
hundredths  of  the  scale,  and  we  can  estimate  to  three  decimals. 

From  the  centre,  Z9  of  the  coordinate  axes  draw  the  circle  ADEC9 
representing  the  earth — the  compression  being  neglected.  The 
north  pole  of  the  earth  is  in  the  plane  YZ,  near  the  point  C,  its 
position  to  be  determined  presently.  The  point  marked  Z  is  the 
axis  of  Z9  perpendicular  to  the  plane  of  the  paper.  The  letter  Z 


37 


ECLIPSE  TABLES. 


55 


ART.  37.     (Continued.) 

A  PORTION  OF  THE  INTERPOLATION  OF  THE  ECLIPSE  TABLES 
FOR  10  MINUTES. 


G.  M.  T.  x.  y.  I.  IL 

6*  0"  —1.57629+0.31835+0.53248—0.01384 
10        1.48334      .28955      .53249  82 


MI-  di.  log  sin  di 

90°  4(X  49"+5°  18'  38//+8.96640 
93    10  54  29  619 


7 

0 

1.01858 

.14552 

.53254— 

.01377 

105 

41 

6   5 

17 

43 

8.96514 

10 

0.92562 

.11671 

55 

76 

108 

11 

9 

33 

493 

20 

.83266+ 

.08789 

56 

75 

110 

41 

12 

24 

472 

8 

0 

.46082— 

.02739 

.53258— 

.01373 

120 

41 

24   5 

16 

47 

8.96387 

10 

.36786 

.05621 

58 

73 

123 

11 

27 

37 

366 

20 

.27490 

.08503 

58 

73 

125 

41 

30 

28 

345 

30 

.18194 

.11386 

58 

72 

128 

11 

32 

19 

324 

40 

—  .08897 

.14269 

59 

72 

130 

41 

36 

10 

303 

50 

+  .00400 

.17152 

59 

72 

133 

11 

39 

1 

282 

9 

0 

.09697 

.20035 

.53259— 

.01372 

135 

41 

42   5 

15 

52 

8.96261 

10 

.18994 

.22918 

59 

72 

138 

11 

45 

43 

240 

20 

.28290 

.25802 

59 

72 

140 

41 

48 

33 

219 

30 

.37586 

.28685 

59 

73 

143 

11 

51 

24 

198 

40 

.46882 

.31569 

58 

73 

145 

41 

54 

15 

177 

50 

.56178 

.34452 

58 

73 

148 

11 

57 

6 

156 

10 

0 

.65474 

.37336 

.53258— 

.01373 

150 

42 

0   5 

14 

56 

8.96135 

10 

.74769 

.40219 

57 

74 

153 

12 

2 

47 

114 

20 

.84064 

.43103 

57 

74 

155 

42 

5 

38 

093 

30 

.93359 

.45986 

56 

75 

158 

12 

8 

28 

072 

40 

1.02654 

.48870 

56 

75 

160 

42 

11 

19 

050 

50 

1.11948 

.51754 

55 

76 

163 

12 

14 

10 

029 

11 

0 

1.21242 

.54638 

.53254— 

.01377 

165 

42 

17   5 

14 

1 

8.96007 

10 

1.30534 

.57522 

53 

78 

168 

12 

20 

13 

51 

.95986 

20 

1.39826 

.60406 

52 

79 

170 

42 

23 

42 

965 

30 

+1.49118—0.63290+0.53251—0.01380 

173 

12 

26  +5 

13 

33 

+8.95944 

also  sometimes  denotes  the  centre  of  this  circle ;  sometimes  the  sur- 
face of  the  earth  above  it ;  and  sometimes  the  point  in  the  celestial 
sphere  which  is  the  zenith  of  the  projection ;  and  in  this  respect  is 
the  point  Z  referred  to  in  CHAUVENET'S  Astronomy,  vol.  i.,  p.  441. 
These  several  points  are  all  in  the  same  straight  line,  and  there  will 
be  no  ambiguity  in  letting  one  letter  represent  them  all.  And  Z 
being  the  zenith,  the  circle  is  the  horizon  at  any  time. 

Next  lay  off  the  values  of  x  along  the  axis  of  X — the  negative 
values  on  the  left,  the  positive  on  the  right,*  because  the  shadow 

*  In  SCHILLEN'S  Spectrum  Analysis,  p.  207,  the  following  unaccountable  collection 
of  mistakes  is  made :  "  As,  however,  the  moon,  which  throws  the  shadow,  only  com- 
pletes its  course  in  the  heavens  round  the  earth  from  west  to  east  in  one  month,  and 
the  earth,  which  receives  the  shadow,  accomplishes  its  revolution  from  west  to  east 
in  one  day,  it  follows  that  the  motion  of  the  moon's  shadow  is  very  much  slower  than 
that  of  the  earth's  surface.  It  therefore  happens  that  the  earth  appears  to  run  away 
from  under  the  moon's  shadow,  or  that  the  moon's  shadow  seems  to  run  over  the 
earth  from  East  to  West." — Translation  by  JANE  and  CAROLINE  LASSELLE.  Edited 
by  WILLIAM  HUGGENS.  D.  Appleton  &  Co. 


56  THEORY  OF  ECLIPSES.  38 

passes  from  west  to  east  over  the  earth ;  on  these  points  erect  the 
ordinates  y,  measuring  the  values  of  y  already  computed,  and  the 
line  connecting  these  latter  points  is  the  path  of  the  centre  of  the 
shadow  across  the  fundamental  plane.  This  line  is  a  curve,  but  of 
very  slight  curvature.  The  hours  are  noted  on  the  path.  The  axis 
of  the  shadow,  which  is  the  line  through  the  centres  of  the  sun  and 
moon,  is  perpendicular  to  the  plane  of  the  paper  at  all  times. 

This  shows  the  beauty  of  the  modern  methods  of  LAGRANGE, 
HANSEN,  and  BESSEL,  that  the  right  ascensions  and  declinations  of 
the  sun  and  moon  are  transformed  into  the  right  ascension  and  decli- 
nation, a  and  d  of  the  point  Z ;  and  from  these  the  coordinate  axes 
are  computed,  so  that  we  now  may  consider  an  apparently  fixed  plane 
with  the  shadow  moving  across  it.  The  earth  can  readily  be  referred 
to  this  plane. 

•  39.  Principles  of  Spherical  Projection. — We  will  digress  for  a  mo- 
ment to  recapitulate  some  of  the  fundamental  principles  applicable 
to  the  orthographic  projection  of  the  sphere.  The  primitive  plane 
of  the  projection  is  here  the  Fundamental  Plane,  and  its  intersec- 
tion with  the  earth,  ADBG,  the  Primitive  Circle,  and  CD,  the  Princi- 
pal Meridian.  All  circles  parallel  to  the  primitive  plane  project 
as  circles  of  their  true  size ;  all  circles  perpendicular  to  the  plane 
project  into  right  lines,  as  the  line  CD.  Circles  oblique  to  the  plane 
project  in  ellipses,  such  as  the  earth's  equator,  which  will  be  projected 
presently.  We  may  revolve  the  whole  sphere  or  only  a  portion  of 
it.  If  we  revolve  it  round  the  axis  Z,  90°,  A  will  fall  at  D,  D  at  B, 
etc.,  and  all  points  of  the  sphere  wrill  revolve  through  the  same  angle, 
though  the  distance  passed  over  is  less  the  nearer  the  point  is  to  the 
axis  of  revolution.  In  any  revolution  all  points  will  revolve  in 
planes  perpendicular  to  the  axis  of  revolution. 

Now  revolve  the  sphere  90°  about  the  line  CD.  The  semicircle 
CAD  and  the  fundamental  plane  will  fall  in  the  right  line  CD.  The 
semicircle  CD  in  space  will  fall  in  the  semicircle  CBD.  In  this 
position  we  know  the  position  of  the  north  pole  and  equator.  The 
point  Z  of  the  celestial  sphere  will  lie  in  the  indefinite  line  Z  T, 
which  in  our  example  has  the  declination  (i=H-5°16/at  the  time 
of  conjunction  ;  therefore,  lay  off  the  angle  BZE'  below  ZB,  so  as 
to  make  the  latter  north  declination,  and  ZEf  will  be  a  portion  of 
the  equator  of  the  earth  projected  in  a  right  line.  It  is  a  principle 
of  this  projection  that  the  elevation  of  the  pole  equals  the  lati- 
tude of  the  point  Z.  Hence  we  may  lay  off  CPf.  Pf  is  the  north 


39 


ECLIPSE   TABLES. 


57 


pole  of  the  earth,  and  P'Z,  a  part  of  the  axis  in  its  revolved 
position. 

When  the  sphere  is  revolved  back,  Pf  will  fall  in  the  line  CD  at 
P,  which  is  the  pole  of  the  earth  in  the  projection.  Er  will  fall  at 
E,  and  the  ellipse  AEB  is  the  equator  projected  obliquely. 

40.  Fig.  3,  Plate  II.— Total  Eclipse  of  I860,  July  18.— This  is  the 
eclipse  CHAUVENET  has  taken  as  his  example,  and  the  projection  is 
made  with  his  data  as  follows  : 

DATA  FOB  FIG.  3,  TOTAL  ECLIPSE,  1860,  JULY  29. 


G.M.T. 

X 

y 

f*i 

E 

Constants  in  Projection. 

0* 

—  1J72 

+  0.917 

358°  31' 

4°  33' 

1 

0.627 

0.757 

13    31 

9    22 

/  +  0.536                IJL\  +  0.262 

T0  2 

—  0.081 

0.596 

28    31 

11     17 

3 

+  0.464 

0.435 

43    31 

19    14 

&!  —  0.009             sin  d  -f  0.358 

4 

1.009 

0.274 

58    31 

24      8 

5 

+  1.554 

+  0.112 

73    31 

28    55 

d  +  20°  57'    n'  sin  d  +  0.094 

1*  mean  time,  sidereal  interval  15*    2m  28s 

Correction  for  equa.  time  and  reduc.  to  a  —    1   31    18 

//,  at  1*  13  31    10 

The  foregoing  remarks,  as  well  as  much  that  follows,  are  applic- 
able to  both  figures,  in  which  the  reference  letters  refer  to  similar 
parts.  The  two  eclipses  differ  from  each  other  considerably  in  some 
respects. 

41.  To  resume  the  consideration  of  the  tables,  x0f  xf  (for  their  differ- 
ence is  so  small  we  cannot  distinguish  between  them  in  the  drawing) 
are  the  spaces  between  the  ordinates  of  the  path  ;  likewise  y0f  and 
yf  are  the  differences  of  the  consecutive  ordinates ;  and  shown  in  the 
right-angled  triangles  at  H,  y'  is  negative  in  each  figure. 

With  the  radius  I  for  Penumbra,  describe  arcs  of  circles  from  each 
of  the  hour  points  on  the  path,  which  are  the  intersections  of  the 
penumbral  cone  with  the  fundamental  plane.  In  Fig.  2  two  of 
these  curves  are  entire,  showing  that  all  the  shadow  falls  upon  the 
earth  ;  in  Fig.  3  it  does  not ;  and  this  circumstance  causes  the  two 
principal  forms  of  the  rising  and  setting  curve,  seen  in  the  Nautical 
Almanac,  and  shown  in  Fig.  17,  Plate  VI.,. and  Fig.  18,  Plate  VIL, 
of  this  work.  The  circles  for  the  umbral  cone  are  too  small  to  be 
shown  on  the  scale  of  Figs.  2  and  3,  and  are  therefore  omitted. 

42.  It  was  stated  in  Article  39  that  the  elevation  of  the  north  pole 


58  THEORY  OF   ECLIPSES.  42 

above  the  fundamental  plane  is  equal  to  the  latitude  of  the  point  Zy 
or  the  declination  of  the  sun.  There  is,  however,  this  important 
distinction  :  If  we  call  tp  the  declination  of  the  sun,  and  d  the  eleva- 
tion of  the  north  pole,  the  compression  of  the  earth  for  these  two 
points  will  not  be  the  same.  The  compression  for  the  angle  of  ele- 
vation d  will  be  the  compression  for  the  latitude  <p  =  90°  —  d. 

To  understand  the  quantities  d  and  log  />,  the  former  must  be  con- 
sidered as  allied  to  the  elevation  of  the  pole  and  with  no  affinity  to 
the  declination  of  the  sun.  In  Fig.  3  (the  elevation  in  Fig.  2  being 
too  small  to  show  this  well)  d  is  the  angle  of  elevation  of  the  pole, 
measured  in  a  vertical  plane ;  but  when  revolved  down  in  the  fun- 
damental plane,  is  the  angle  P'ZP. 

It  should  first  be  remarked  that  analytical  geometry  gives  the 
minor  axis  of  an  ellipse,  b  =  aV\  —  e2,  e  being  the  eccentricity ;  and 
in  the  present  case  a,  the  equatored  radius  of  the  earth,  and  major 
axis,  is  taken  as  unity.  We  also  learn  that  if  a  circle  is  described 
upon  the  "  major  axis  of  an  ellipse,  all  the  ordinates  of  the  ellipse 
will  be  shortened  from  those  of  the  circle  in  the  same  ratio  as  the 
minor  axis  "  ;  that  is,  in  the  ratio  1/1  —  e2. 

Now  sines  and  cosines  are  always  given  to  a  radius  unity ;  when 
cos  d  is  multiplied  by  this  quantity  it  will  very  nearly  represent  the 
cosine  of  the  arc  of  the  sphere  at  C.  CZ,  it  will  be  remembered,  is 
less  than  unity  on  account  of  the  compression,  and  P'Z  is  still  less 
on  account  of  the  compression  being  greatest  at  the  pole. 

If  we  take  the  sum  of  the  squares  of  equation  (39),  which  gives 
log  plt  we  get,  after  reducing,  pl  =  1/1  —  e2  cos2  d.  The  general 
expression  for  the  radius  of  the  earth  is, 


1  —  2e2  sin2  <f>  -f  e*  sin2  <p 
1  —  e2  sin2  <p 

in  which,  if  we  make  a  small  approximation,  writing  sin4  tp  for  sin2  tp 
in  the  last  term  of  the  numerator,  the  latter  becomes  a  perfect  square, 
and  we  have 

p  =  I/I  —  e2  sin2  ^. 

This  becomes  the  same  precisely  as  the  former  equation,  if  we  make 
<p  =  90  —  d,  or,  what  is  the  same  thing,  write  cos  d  for  sin  tp,  so  that 
the  quantity  p1  is  very  nearly  represented  by  the  line  CZ  (Fig.  3). 

We  can  now  understand  the  office  which  log  pl  fulfils.  It  results 
directly  from  CHAUVEKET'S  transformation  of  the  geographical  lati- 
tude tp  through  the  geocentric  tp'  to  the  transformed  latitude  tp^  pl 


42  ECLIPSE   TABLES.  59 

appears  as  a  divisor  to  the  coordinate  37,  which  is  placed  equal  to  ^  j 
thus, 

*-•*• 

Pi 
and  f !  assumed,  so  that 

£  +  ^1+^  =  unity, 

the  meaning  of  which  is  that  the  earth's  spheroid  is  replaced  by  a 
sphere  (radius  unity)  which  lies  outside  of  the  earth  at  the  poles, 
because  r]l  >  y  •  and  probably  within  the  earth  at  the  equator.  And 
in  order  to  retain  the  proper  distance  between  the  earth  and  shadow 
path,  its  ordinate  y  is  likewise  divided  by  pl9  giving  yr 

43.  The  Greenwich  hour  angle  of  the  principal  meridian  PD  is 
fj.ly  which  is  shown  clearly  thus  : 

/ij  =  S  -f  h'  —  a  =  0  +  hf  —  («'  -f  small  term,  equa.  17). 
But 

S  —  a'  =  —  equation  of  time. 
Hence, 

fil=  h  —  equa.  time  —  small  term.  (52) 

As  the  hour  angle  is  counted  from  apparent  noon,  we  must  reduce 
from  mean  noon  to  apparent  by  applying  the  equation  of  time  accord- 
ing to  its  sign  on  the  mean  time  page  of  the  Almanac,  which  is  its 
negative  value.  The  small  term  reduces  the  angle  from  the  sun's 
right  ascension  to  the  quantity  a.  We  will  compute  this  for  9*  by 
both  formulae  for  comparison  : 


B  168°   6'  14".70 

h  135     0     0 

Keduc.tosid.     0   22  10  .62 


303   28   25  .32 
167  46  43  .49 


a'  167°  46'  44".34 

8 -a'  0  19  30   .36 


—  Equa.  time  +  2OT  46'.73 

—  "        "    arc  0°  41'  40".95 
h  mean  time                 135      0     0 
Small  term  —  (—  0.86)  -f  0  .86 
Mi                                  135    41   41  .81 


135  41   41   .83  , 

Reduc.  to  sid.  0°  22'  10".62 


0-a'  0    19   30  .36 

Equa.  time  0    41   40   .98 


It  is  seen  that  the  equation  of  time  takes  the  place  of  the  two 
smaller  terms,  being  just  equal  to  them. 

In  the  first  formula  hr  is  in  sidereal  time,  and  in  the  second  formula 
h  is  in  mean  time. 

Now  suppose  in  Fig.  2  the  earth  be  revolved  so  that  the  north 
pole,  P,  falls  in  the  point  Z,  the  circle  AD  EC  will  be  the  equator 
and  ZD  the  principal  meridian  at  all  times  while  the  earth  revolves. 
We  will  now  set  off  from  D,  the  hour  angle  of  9A—  135°,  toward  the 


60  THEORY   OF   ECLIPSES.  43 

right,  to  the  point  g,  which  is  Greenwich  mean  noon  ;  but  the  merid- 
ian of  Greenwich  is  not  yet  overhead  until  we  further  set  off  the 
equation  of  time  to  Greenwich  apparent  noon,  G.  This  and  the  small 
reduction  to  a,  the  meridian  of  the  point  Z,  gives  us  the  hour  angle 
fa,  between  the  Greenwich  meridian  and  the  principal  meridian, 
PZD.  We  may  likewise  set  off  the  other  hours  from  6  to  12,  which 
give  the  hour  angles  at  these  hours. 

44.  Lastly,  we  have  the  quantities  6',  c',  E,  and  e  to  illustrate  ; 
they  are  closely  related. 

If  we  omit  the  small  terms,  we  have  — 

tan£=-=  (53) 


As  E  is  counted  from  the  axis  Y,  set  off  xf  from  Z  (Fig.  2)  along 
the  axis,  and  —  y*  (which  is  positive)  on  the  right  to  the  point  F. 
This  value  of  E  gives  a  line  at  right  angles  to  the  path.  The  small 
terms  fi\  x  sin  d  and  fi\  y  sin  d  may  be  represented  in  the  following 
manner  :  p\  in  natural  numbers  is  0.262,  and  sin  d  0.091  ;  and  x, 
being  multiplied  by  these,  is  reduced  by  the  amount  of  their  product, 
0.024  ;  so  that  x  and  y,  if  drawn  on  a  reduced  scale  of  0.024,  will  repre- 
sent these  small  terms.  This  is  too  small  in  Fig.  2  to  be  shown  on  ac- 
count of  the  small  declination,  but  in  Fig.  3  we  have  from  the  data,  Art. 
40,  p.\  0.262,  sin  d  0.358  and  their  product,  0.094,  the  reduction  for 
the  miniature  representation  of  the  path.  Now  transfer  this  figure 
parallel  to  itself,  so  that  the  point  Z  is  placed  upon  F,  and  number 
the  hours.  By  this  means  we  add  the  small  terms  to  the  large  one, 
giving  &',  the  several  distances  of  the  points  1,  2,  3,  etc.,  from  the 
axis  CD  ;  and  c'  their  distances  from  the  axis  AB.  We  also  have 
the  angle  E  at  1*,  1  ZC,  and  e  the  distance  1  Z. 

E  at  the  middle  of  the  eclipse  gives  a  line  very  nearly  at  right 
angles  to  the  path  of  the  shadow.  We  shall  recur  again  to  this  angle 
under  Art.  97,  on  the  Northern  and  Southern  Limiting  Curves. 

45.  Motion  of  the  Successive  Eclipses  of  a  Series.  —  Applying  the 
Criterion  in  Art.  12  to  the  eclipse  of  1904,  September  9,  we  see  that 
it  is  at  the  moon's  ascending  node,  and  the  series  is  therefore  moving 
south.  As  the  eclipse  at  local  apparent  noon,  the  point  where  the 
path  crosses  the  axis  of  F,  is  south  of  the  point  Z,  the  successive 
eclipses  are  decreasing.  In  Fig.  2,  Plate  I.,  the  path  of  the  preced- 
ing eclipse  of  1886  is  shown.  The  eclipse  of  1860,  July  29  (Fig.  3, 
Plate  II.),  CHATJVENET'S  example,  is  at  the  moon's  descending  node, 
the  series  moving  north,  and  successive  eclipses  decreasing.  The 


45  ECLIPSE   TABLES.  61 

paths  of  the  two  following  eclipses,  1878  and  1896,  are  also  shown 
in  this  figure.  And  here  we  notice  another  point,  the  inclination  of 
the  path  is  changed  at  each  appearance.  When  the  angle  gets  to  its 
maximum,  it  will  then  decrease,  and  in  time  the  path,  instead  of 
moving  south  across  the  earth,  will  move  north. 

To  ascertain  whether  the  inclination  of  the  path  will  increase  or 
decrease  for  the  next  eclipse,  we  have  as  follows  :  The  Saros  (Art.  1 1) 
is  18y  10  or  lld  7A  42m.  Our  year  is  the  period  of  the  sun's  revo- 
lution, and  the  time  is  sure  to  be  longer  than  the  year  by  10  or  11 
days,  in  which  time  the  sun's  longitude  has  increased  some  10  degrees, 
and  its  right  ascension  between  the  limits  of  36m  and  44m.  As  the 
sun  and  moon  will  be  together  at  the  next  eclipse,  the  moon's  right 
ascension  must  also  increase  by  the  same  amount.  Taking  the  eclipse 
Sept.  9,  1904,  9A  the  time  of  conjunction,  we  have  from  the  moon's 
ephemeris  for  this  date  and  hour  the 

Change  between  9  and  10  hours,      -f  2m  26*  —  11'  34" 

And  on  page  146  for  the  sun,          9      and     — 57 

137  —       637 

The  angle  of  the  path  will  be  637  divided  by  137,  when  the  latter  is 
reduced  to  arc  and  multiplied  by  the  moon's  declination,  but  as  it  is 
only  the  relative  values  of  the  angle  we  want,  we  may  dispense  with 
this  reduction,  and  the  quotient  of  the  above  is  46.5. 

Next  take  the  sun's  right  ascension  ten  days  later,  llh  45m  40'. 
In  the  moon's  ephemeris  this  value  occurs  on  the  9th  at  23* : 
The  hourly  motions  at  the  time  are  2  25  —  11  52 

Sun's  hourly  motions  Sept.  19,  9^  —        58 

Difference,  136  654 

The  quotient  is  48.1 

Comparing  this  with  the  previous  quotient,  we  see  that  the  inclina- 
tion is  slightly  increasing.  The  method  is  only  a  rough  approxima- 
tion, but  it  is  sufficient  for  the  purposes  for  which  it  is  intended,  in 
connection  with  Table  XV.  at  the  end  of  this  work.  We  see  in 
Fig.  2,  Plate  I.,  that  the  inclination  has  increased  in  1904  since  1886. 
The  motion  of  the  successive  eclipses  north  or  south  must  be  esti- 
mated on  the  axis  of  F,  for  this  is  the  time  of  conjunction.  In  Fig.  3, 
Plate  II.,  the  total  eclipses  of  1860,  1878,  1896,  it  is  seen  that  after 
two  more  appearances,  the  central  path  will  fail  to  touch  the  earth, 
and  the  eclipse  will  be  partial.  About  twelve  partial  eclipses  will 
then  occur,  when  the  shadow  will  fail  to  touch  the  earth  and  the 
series  will  have  run  out.  The  series  commenced  at  the  south  pole, 
followed  by  12  or  13  partial  eclipses;  then  40  or  50  total,  the 


62 


THEORY   OF   ECLIPSES. 


45 


whole  series  of  any  eclipse,  Professor  NEWCOMB  states,  occupies 
about  1000  or  1200  years,  during  which  there  are  60  or  70  eclipses. 
The  series  of  1860,  it  is  seen,  passed  over  about  0.10  of  the  earth's 
radius  in  18  years,  which  is  greater  than  usual.  The  eclipse  of  1883- 
1901,  May,  passed  over  0.08,  while  that  of  1883-1901,  October, 
passed  over  but  0.03.  The  motion  is  variable  in  one  series.  Our 
example,  1904,  September  9,  has  passed  over  since  1886,  0.06,  as 
shown  in  Fig.  2. 

46.  The  Umbral  Cone  and  Species  of  Eclipse. — If  in  equation  26 
we  omit  the  small  term  and  place  the  cosines  of  small  angles  equal 
to  unity,  we  have 


sin  TT 

and  with  the  values  of  the  moon's  parallax  given  in  Art.  10,  we 
have  the  greatest,  least,  and  mean  values  of  the  moon's  distance,  z, 
as  given  in  the  table  below. 

The  greatest,  least,  and  mean  values  of  the  sun's  distance,  as  given 
by  its  logarithms  in  NEWCOMB'S  tables  of  the  sun,  and  corrected  for 
the  constant  0.0000460  there  deducted,  are 

0.0072147  9.9926636  0.0000000 

Computing  the  umbral  cone  for  these  four  maximum  and  minimum 
values,  there  results  the  several  quantities  given  in  the  subjoined  table : 

THE  UMBRAL,  CONE. 


Moon's 
Parallax. 

SUN'S  D 
Greatest, 
log  r'  0.00721 

ISTANCE. 

Least, 
log  r'  =  9.99266 

Greatest. 
61'  28.8 

z  = 
k 

55.92 
59.50 

—  3.58 
-  0.016 

z 
k 

55.92 
57.54 

-1.62 
—  0.008 

sin  / 
c 

i>  ' 

sin  / 
I, 

Least. 
53'  55".0 

z 
Ic 

63.76 
59.50 

+  4.26 
+  0.019 

z 
k 

63.76 
57.54 

+  6.22 
+  0.029 

sin  / 
c 
k 

sin  / 
I 

Mean. 
57'  2.55 

z 

60.27 

z 

60.27 

46  ECLIPSE   TABLES.  63 

We  will  now  plot  these  quantities  in  a  diagram  (Fig.  4,  Plate 
III.)  to  a  vertical  scale  of  half  an  inch  to  one  unit,  which  is  the 
earth's  radius,  and  assume  the  horizontal  line  AB  as  the  mean 
fundamental  plane  to  which  all  other  quantities  will  be  referred. 
The  distance  z  is  the  distance  of  the  fundamental  plane  from  the 
moon.  Drawing  lines  for  the  greatest  and  least  distances  and  also 
computing  these  points  for  each  minute,  we  have  on  the  left  of  the 
figure  the  quantities  depending  upon  the  moon's  distance.  This  deter- 
mines the  position  of  the  fundamental  plane,  which  depends  chiefly 
upon  the  moon's  parallax. 

k 

The  values  of computed  for  the  sun's  greatest  distance  are 

sin/ 

very  nearly  the  same — 59.513  and  59.494  ;  and  for  the  sun's  least  dis- 
tance, 57.548  and  57.529,  differing  only  two  units  in  the  second  deci- 
mal. This  shows  that  the  moon's  parallax  has  but  little  effect  in 
determining  the  position  of  the  vertex  of  the  umbral  cone.  Mean 
values  are,  however,  given  in  the  above  table,  and  thence  c,  by  which 
the  vertex  of  the  cone  can  be  located  thus :  The  quantity  c  is  3.58 
below  the  upper  extreme  plane,  and  4.26  above  the  lower  extreme; 
their  numerical  sum  is  7.84,  the  distance  of  the  two  extreme  planes 
apart.  Likewise,  for  the  sun's  least  distance,  we  have  — 1.62  and 
+  6.22,  whose  numerical  sum  is  also  the  same.  These  give  the 
vertices  of  the  umbral  cone  in  its  extreme  positions,  which  points 
we  can  set  off,  drawing  horizontal  lines  from  them  to  the  right 
side  of  the  drawing.  Between  these  two  latter  lines,  if  we 
describe  a  circle,  the  tangent  point  to  the  lower  line,  the  sun's 
least  distance  may  be  taken  as  the  point  of  solar  perigee  at 
(nearly)  January  1 ;  the  upper  tangent  point  will  be  the  apogee, 
July  1,  and  pointing  off  the  months  on  this  little  circle  we  may  thus 
very  nearly  ascertain  the  position  of  the  vertex  of  the  umbral  cone 
at  any  time  of  the  year ;  for  it  is  almost  independent  of  the  moon's 
parallax.  The  moon's  parallax  being  given,  we  can  at  any  time  by 
it  locate  the  position  of  the  fundamental  plane.  Just  above  the 
vertex  of  the  cone  on  the  line  representing  the  moon's  greatest  par- 
allax lay  off  the  radii  of  the  umbral  cone  —  0.016  and  —0.008. 
Likewise  on  the  line  of  least  parallax  lay  off  the  other  values  of  the 
umbral  cone,  -f  0.019  and  -f  0.029.  Lines  from  these  points  to  the 
two  vertices  already  located  will  form  the  elements  of  two  cones 
superimposed ;  and  the  points  taken  three  and  three  will  be  found 
to  be  nearly  in  the  same  straight  line.  These  values,  which  are 


64  THEORY  OF   ECLIPSES.  46 

radii  of  the  cones  are  here  laid  out  as  diameters,  so  as  to  enlarge  the 
scale  of  measurement. 

In  the  present  eclipse  the  data  September  9  and  parallax  61'  23" 
give  the  radius  of  shadow  for  total  eclipse,  0.0137,  which  is  sub- 
stantially correct,  the  construction  being  shown  in  dotted  lines  as 
follows  :  From  the  given  date,  September  9,  draw  the  horizontal  line 
to  Vj  which  is  the  vertex  of  the  cone ;  draw  the  other  element,  ve. 
Then  draw  from  the  given  value  of  the  moon's  parallax,  61 '  23",  the 
horizontal  line  ae,  which  will  represent  the  Fundamental  Plane  and 
its  intersection,  be,  with  the  cone,  is  the  radius  of  the  shadow,  which 
can  be  measured  by  scale.  This  figure  may  be  used  for  any  eclipse. 

CHAUVENET'S  example,  July  29,  parallax  59'  48 ",  gives  also  a 
total  eclipse  and  radius  of  cone,  0.009. 

47.  But  the  diagram  tells  us  much  more  than  this.     We  see  that 
as  the  mean  fundamental  plane  lies  in  the  shadow  of  an  annular 
eclipse,  the  shadows  of  these  must  be  generally  greater  in  diameter 
than  in  total  eclipses ;  which  is  true.     Also,  as  the  moon  vibrates 
between  the  extreme  parallaxes,  if  it  is  near  one  limit  at  one  eclipse 
it  will  very  likely  be  near  the  other  limit  for  the  other  eclipse  six 
months  after  ;  and  both  eclipses  will  be  large.     This  occurs  notably 
in  1901,  1904,  etc.     If,  on  the  other  hand,  the  moon  is  not  near  its 
limit  at  one  eclipse,  it  will  very  likely  not  be  near  either  limit  six 
months  after,  and  both  eclipses  will  be  small.     It  may  happen  that 
in  both  cases  the  moon  is  near  the  mean  parallax,  and  two  annular 
eclipses  will  occur,  which  was  the  case  in  1897,  when  there  were  only 
two  eclipses,  both   solar  and  both  annular.      Sometimes  when  the 
sun  is  not  near  the  node  there  may  be  two  partial  eclipses  at  one 
node  and  one  at  the  other  node.     In  1888  there  were  three  partial 
eclipses,  all  of  the  annular  species,  and  no  total  solar  that  year. 

If  in  the  mean  fundamental  plane  we  draw  a  semicircle  repre- 
senting the  earth's  hemisphere,  we  see  that  the  vertex  of  the  lower 
cone  is  within  the  earth's  surface.  In  this  case,  as  the  shadow  moves 
over  the  earth,  at  the  beginning  and  end,  the  eclipse  will  be  annular 
and  total  in  the  middle.  This  occurred  in  1890,  December  11,  and 
it  is  properly  called  a  Total-Annular  Eclipse.  It  should  be  classed 
as  Annular  because  ^  for  umbra  has  the  positive  sign.  The  moon's 
parallax  for  this  eclipse  is  59r  8. "2  at  conjunction. 

48.  From  this  diagram  it  seems  at  first  glance  as  though  we  might 
approximate  in  numbers  to  the  relative  frequency  of  total  and  annu- 


48  ECLIPSE   TABLES.  65 

lar  eclipses ;  but  I  do  not  think  it  can  be  done.  The  vertex  of  the 
cone  moves  quite  regularly  between  its  extremes ;  but  the  moon 
does  not.  Its  motion  as  to  its  parallax  is  very  irregular,  seldom 
reaching  the  extremes  here  given.  Another  fact  which  would  vitiate 
any  computation  is  that  at  either  node  there  may  be  one  or  two  solar 
eclipses.  After  a  series  of  years  in  which  there  is  but  one,  another 
series  may  enter,  making  two.  The  best  method  of  arriving  at  the 
proportion  is  to  count  the  number  of  eclipses  in  one  Saros,  counting 
the  partial  eclipses  as  total  or  annular,  according  to  the  sign  of  llt 
Total-Annular  Eclipses  should  be  classed  as  Annular  on  account 
of  the  sign  of  ^  in  the  Fundamental  Plane.  And  here  is  a  third 
circumstance  which  would  again  vitiate  any  computation — a  total 
eclipse  may  after  lapse  of  years  become  annular,  and  vice  versd. 

49.  This  fact  does  not  seem  to  be  stated  in  any  general  Astronomy, 
that  the  species  of  total  or  annular  may  change.  In  this  connection 
the  following  successive  eclipses  of  one  series  are  interesting ;  the 
semidiaraeters  only  are  given  : 

Sun's  Moon's 

.  Semidiameter.         Semidiameter. 
1800,  Interpolated.    Brit.  N.  A.  16'    7".3  ,  -,  ,     16'  18".   —  1 

1818,       "  "        "  16     8   .7      21     16   17   '   ~~2'4 

1836,  Nov.  8,  Total  Eclipse.    Brit.  N.  A.       16  10  .8      ^     16  14  .6  —  2.5 

1854)No,19,TotalEc,ipS,      «        -  »»'  +  «{££;*£££.. 

1872,  Dec.  29,  Annular  Eclipse,  Amer.  Eph.    16   16  .0  _  l  Q     16     8   .7  —  6.0 
1890,  Dec.  11,  Total  Annular.  "  16   15   .0  16     6  .1      2.6 

1908,  1927,  Total  Annular.     OPPOLZEB,  Canon  of  Eclipse. 
1945,  Annular  Eclipse.  "  "  " 

The  change  in  the  English  Nautical  Almanac  of  1854,  increasing 
BTJRCKHARDT'S  value  of  the  moon's  Semidiameter  by  2".60,  it  is  cu- 
rious to  note,  changed  the  species  of  the  eclipse.  With  the  old  value  the 
eclipse  would  have  been  Total -Annular.  In  1872  it  seems  doubtful 
whether  the  true  character  of  the  eclipse  was  known  to  the  com- 
puter, since  the  limits  of  the  umbral  cone  were  not  required  in  those 
days,  and  hence  the  change  in  the  sign  of  L  on  the  earth's  surface 
escaped  notice.  It  is  classed  as  an  Annular  Eclipse  in  the  American 
Ephemeris. 


66  THEORY  OF   ECLIPSES.  50 

SECTION    Y. 

EXTREME  TIMES  GENERALLY. 

50.  WITH  this  section  commences  the  work  for  the  eclipse  itself. 
Where  figures  are  to  be  given,  the  computation  should  be  made  with 
five-place  logarithms  to  seconds  of  arc  ;  but  where  no  figures  are  to 
be  given,  four-place  logarithms  to  minutes  of  arc  are  sufficient,  for 
these  quantities  are  required  only  for  plotting  oil  the  chart. 

CHAUVENET'S  formulae  for  this  section  are  as  follows  : 

M0  sin  MO  =  x0 
M0  cos  MQ  —  yQ 

».«.£-«/>  (55) 

n  cos  N  =  y0f  ) 

(5g) 


P  ~r  I 

r  =  £-±1  cos  ^  -  —  °  cos  (M0  -  N)  (57) 
n                     ft 

T=T0+T  (58) 

r=N+<p  (59) 

tan  Y'  =  Pi  tan  r  (60) 


=l 
sm  Y  cos 


(62) 
tan  rr  =  Pi  tan  r  (63) 

(64) 


cos  <f>i  sin  i9  ==  sin  Y'  ^ 

cos  ^  cos  #  =  —  cos  /  sin  e^  >  (65) 

sin  ^!  =  cos  ^'  cos  c?j      J 

a>  =  ^  _  «  (66) 

Use  five-place  logarithms  and  to  seconds  throughout.  Compute 
log  mQ  and  M0  once  for  the  epoch  hour,  then  two  approximations  are 
necessary  for  the  times.  In  the  first,  take  the  quantities  for  the  epoch 
hour  and  p,  which  is  unknown,  equal  to  unity.  There  will  be  but 
one  column  for  this  ;  but  when  sin  <j)  is  reached  there  will  be  two 
angles  resulting  :  one  greater  than  90°  with  its  cosine  negative,  and 
the  other  less  than  90°  with  its  cosine  positive.  The  angles  may  be 
either  +  or  —  ;  and  the  double  sign  of  cos  <p  will  give  two  values 


50  EXTREME  TIMES  GENERALLY.  67 

for  the  times,  which  may  be  placed  below  one  another  in  the  same 
column.  Log  n  is  not  needed  in  the  numerator,  so  that  log  -  may 

be  gotten  instead. 

For  the  second  approximation  compute  in  two  columns,  taking  out 
in  each  the  several  quantities  for  the  times,  Ty  already  found  and  com- 
pute the  quantity,  p,  for  beginning  and  for  ending.  Then  with  a?0'  and 
y0'  compute  n  and  N  for  these  times,  and  proceed  with  the  formulae 
for  T,  the  final  times.  For  beginning,  <p  is  taken  as  >  90°  with  its 
cosine  negative,  and  for  ending  <  90°  with  its  cosine  positive.  The 
quantity  sin  <p  for  beginning,  indeed,  gives  two  values,  but  the 
acute  value  for  ending  will  be  less  accurate  than  the  first  approxi- 
mation, for  the  quantities  from  which  it  is  derived  are  taken  further 
from  the  time  of  ending  than  in  the  first  approximation.  Similar 
remarks  apply  to  the  ending.  These  other  values  could  be  used  as 
a  check,  but  as  the  final  times  agree  so  nearly  with  the  first  approxi- 
mation, this  is  a  partial  check  in  itself. 

In  the  second  approximation  at  least  four  decimals  of  a  minute 
are  required  to  get  the  longitude  correctly,  so  that  the  terms  should 
be  carried  out  to  five  decimals,  though  only  one  is  given  in  the 
Almanac. 

Having  the  final  times,  omit  Nos.  59,  60,  61,  and  proceed  with 
62.  7-  and  f'  differ  very  slightly,  so  that  the  signs  of  sin  f  and  cos  f 
are  readily  known  from  f.  Sin  dl  is  interpolated  in  the  tables  and 
also  in  our  example  to  every  10  minutes,  and  should  be  still  further 
interpolated  to  the  times  T;  cos  d^  is  given  for  every  hour.  The 
first  and  second  of  No.  64  both  give  cos  <pt  and  I  always  get  both 
values ;  for  if  there  is  any  inconsistency  in  the  above  work,  it  will 
show  itself,  but  this  will  not  detect  other  errors.  And  also  sin  <p^ 
and  cos  ^  should  give  the  same  angle  within  one  unit  of  the  last 
decimal  of  logarithms.  By  consistency,  I  mean  that  all  the  sines 
and  cosines  used  in  these  equations  must  belong  to  the  same  angle ; 

and  the  additions  of  the  logarithms  must  be  correct.     Log 

1/1  —e2 

is  found  among  the  constants  (Art.  23).  &  is  the  local  apparent 
time  of  the  phenomena  throughout  this  discussion,  and  can  be  re- 
duced to  mean  time  by  applying  the  equation  of  time,  fa  must  be 
taken  from  the  eclipse  tables  for  the  10m  preceding  the  time,  the 
proportional  parts  for  the  time,  T,  can  be  taken  from  Table  VIII. 
and  added  to  it.  These  values  for  beginning  and  ending  may  be 
checked  by  their  difference  agreeing  with  the  change  of  /JtL  for  the 


68       .  THEORY  OF   ECLIPSES.  50 

difference  of  the  times.     The  larger  term  of  the  difference  can  be 
taken  from  the  tables,  and  the  proportional  part  from  Table  VIII. 

p  is  the  radius  of  the  earth  for  the  place  of  contact  and  may  be 
checked  by  Table  IV.,  taking  <p  as  the  argument.  The  logarithms 
should  agree  within  one  unit,  or  at  most  two  units,  of  the  last  place 
of  decimals.  T  results  in  decimals  of  an  hour,  which  can  be  reduced 
to  minutes ;  but  the  decimals  are  used  as  the  argument  for  /^,  in 
Table  VIII.  It  is  preferable  and  much  easier  to  take  $  as  minus 
for  beginning  and  plus  for  ending,  and  greater  or  less  than  90°  ;  the 
angle  will  then  have  the  same  sign  as  its  sine. 

51.  Generally  throughout  this  discussion  angles  that  are  measured 
from  the  principal  meridian,  such  as  M,  f,  (p   in  the  Maximum,  are 
<  90°  in  the  northern  hemisphere — that  is,  north  of  the  point  Z,  and 
>  90°  in  the  southern.     &  in  this  respect  depends  upon  the  eleva- 
tion of  the  pole.  And  they,  with  also  $,  if  taken  less  than  180°  and 
either  +  or  —  are  always  —  west  of  the  axis  of  Yy  and  always  + 
when  east.     So,  if  they  are  run  around  from  0°  to  360°,  the  reader 
will  be  likely  not  to  see  this  symmetry.     The  angle  (p  used  in  this 
section  is,  however,  quite  a  different  angle  from  $  in  the  Maximum 
Curve  alluded  to  a  few  lines  above. 

52.  Middle,  of  the  Eclipse. — CHAUVENET  does  not  give  this  for  the 
eclipse  generally,  but  it  may  be  found  thus — on  page  469,  vol.  i., 
of  his  Astronomy,  is  the  following  pair  of  equations  (not  numbered), 
which  he  develops  into  the  formulae  for  extreme  times : 

(p  +/)  sin  (M  —  N}  =  m  sin  (Jf0  —  JV) 
(p  +  0  cos  (M—  N}  =  w0  cos  (Jf0  —  N)  +  nr 
If  we  take  the  sum  of  the  squares  of  these  we  have 

(p  +  iy  =  m02  sin2  (M0  —  JV)  +  [m0  cos  (Jf0  -  JV)  +  Mr]2     (67) 

Now  as  the  shadow  approaches,  m  diminishes  until  at  its  least 
value  the  eclipse  is  evidently  the  greatest ;  m  then  increases  and  the 
eclipse  becomes  less. 

In  the  above  equation  MQ  and  m0  are  taken  at  the  epoch  hour ; 
N  varies  slowly,  and  is  sensibly  constant  for  small  intervals  of  time  ; 
the  last  term  is  essentially  positive,  being  a  square  ;  and  m  =  p  -{-  I 
is,  therefore,  a  minimum  when  the  last  term  is  zero,  whence  we  have 

T=_^cos(j/0_j\r)  (68) 

n 
And  for  the  time, 

T=T0  +  r.  (69) 


52  EXTREME  TIMES  GENERALLY.  69 

This  term  is  already  given  in  the  preceding  computation,  and  the 
mean  of  the  two  values  for  beginning  and  ending  should  be  taken. 
The  time  of  greatest  eclipse  is  considered  as  the  Middle  of  the  Eclipse 
as  it  results  from  the  variable  quantities  taken  at  the  middle  time. 
It  is  usually  not  equidistant  between  the  two  extremes,  but  it  is  con- 
sistent with  the  principle  that  the  quantities  for  computing  a  phe- 
nomenon should  be  taken  near  the  time  of  that  phenomenon.  When 
the  path  is  much  inclined,  the  compression  of  the  earth  will  be  differ- 
ent for  the  two  points  of  contact,  and  this  will  affect  the  times  differ- 
ently. No  figures  are  given  for  this  time  in  the  Almanac  unless  the 
eclipse  is  partial,  to  which  the  reader  is  referred,  Art.  78,  Section 
VII.,  on  Maximum  Curve. 

The  computer  will  find  that  the  Middle,  as  given  by  the  extreme 
times  of  Central  eclipse  and  of  Limits  of  Penumbra,  each  give 
slightly  different  values. 

53.  Criterion  for  Partial  Eclipse. — This  is  obtained  from  the  com- 
putation of  extreme  times  generally. 

m  sin  (If—  N)>p.  (70) 

This  expression  gives  the  distance  of  the  path  from  the  centre 
sphere,  and  is  already  computed,  p  is  the  radius  of  the  earth  at 
this  point,  which  may  vary  from  log  p  at  90°  =  9.9985  to  9.9987  at 
70°  (Table  IV.). 

54.  Example  of  the  Computation. — This  is  given  just  as  it  was 
made,  except  that  the  first  approximation  is  omitted.     It  is  in  one 
column  with  double  values  after  getting  sin  ^,  one  placed  below  the 
other  to  economize  space,     p  -f-  I  is  taken  as  1  -f- 1. 

The  times  from  the  first  approximation  and  the  angle  ^,  which 
are  needed  in  the  rest  of  the  work,  are  placed  after  m0.  Just  before 
getting  the  final  times  the  figures  (1)  (2)  denote  the  two  terms  of 
equation  (57).  The  proportional  parts  for  fa  are  given  in  this  exam- 
ple, among  them  the  small  term  +  2".  The  table  of  proportional 
parts,  which  is  based  on  a  change  of  fa  =  1.5°  Om  0s  in  one  hour, 
but  this  varies  slightly — sometimes  more,  sometimes  less.  In  this 
example  (see  Tables)  it  is  about  18"  in  one,  or  3'  in  ten  minutes. 
The  time  of  beginning  is  6*  7m.8,  and  the  change  in  this  7m.8,  which 
the  proportional  parts  do  not  take  account  of  is  the  -j-  2",  mentioned 
above.  For  ending  this  correction  is  zero. 


70  THEORY   OF   ECLIPSES.  54 

EXTREME  TIMES  GENERALLY,  TOTAL  ECLIPSE,  1904,  SEP- 
TEMBER 9. 


(54)  x0             +  8.98664 

(62)  y           +  281     7  48 

+  113   19  46 

y0               -  9.30179 

(63)  tany 

0.70611 

0.36524 

tan  M}           9.68485 

tany' 

0.70465 

0.36378 

M0      +  154   10  22 

sin  y' 

-  9.99170 

+  9.96273 

sin                  9.63914 

cosy7         +  9.28705 

—  9.59894 

9.34750 

(64)  sin  d,          +  8.96623 

+  8.95963 

cos                 9.95429 

COS  0  COS  $   - 

-  8.25328 

+  8.55857 

Iogm0       +  9.34750 

tantf 

1.73843 

1.40416 

(55-58)  from  first  approximation.                          # 

91     2  47 

+  87    44  31 

T                 6    7.806 

11  20.958 

sin 

9.99992 

9.99966 

V         +  173  54   30 

+  6    5   30 

9.99178 

9.96307 

cos 

8.26151 

8.595,50 

(59)  y         +  281    8   26 

113   19  26 

cos^          +  9.99177 

+  9.96307 

(60)  sin  y               9.99174 

9.96297 

cose/!         +  9.99813 

+  9.99813 

tan  y               0.70570 

0.36536 

sin  0t 

f  9.28518 

-  9.59707 

tan  Y              0.70424 

0.36391 

tan^         +  9.29341 

—  9.63400 

(61)  sin  y'              9.99168 

9.96275 

(65)tan^          +  9.29488 

-  9.63547 

p              +  9.99994 

+  9.99978 

0            + 

11     9  18 

-  23   21  50 

I                +  9.72631 

9.72634 

I  —  I  A           0.27363 

0.27344 

(66)  ft  Tables 

90   40  49 

170   42  23 

.£                   0.45905 

0.45893 

pp 

1   45 

0   13  30 

log  (p  +  0+0.18536 

+  0.18527 

12    0 

9 

Table  VIII.  • 

18 

8 

(55)  V             +  9.74644 

+  9.74640 

6 

0 

y/                 -    .23780 

—  9.23810 

.-       +  2 

tan  N               .50864 

9.50830 

Pi 

92   38  15 

170   56  10 

JV        +  107    13  24 

+  107    14  10 

sin                  9.98007 
log  n               9.76637 

9.98004 
9.76636 

(66).       |+183   41     2 
I—  176    18  58 

+  83   11  39 
=  east. 

cos                  9.47143 

9.47175 

log  1  :  n     +  0.23363 

+  0.23365 

(56)  3/—  2V  +  46   56  58 

+  46   56  12 

Middle. 

sin(    •)      +  9.86377 

+  9.86368 

cos  (    )      +  9.83419 

+  9.83430 

msin          +  9.21127 

+  9.21118 

(69)  T-=  (2)  mean  value 

—  0.260 

sinV          +  9.02591 

+  9.02591 

(70)  T 

8.740 

V»          +  173   54  24 

+  6     5  36 

8*  44.40 

(57)  cos  V         —  9.99754 

+  9.99754 

log(l)       —0.41653 

+  0.41646 

(2)             +  9.41535 

+  9.41545 

Nos.  (1)      -  2.60933 

+  2.60894 

—  (2)    •-  0.26022 

-  0.26029 

r                 -  2.86955 

+  2.34865 

(58)  T  I                6'13045 

11.34865 

v     '       1             6»7m.8270       11*  20TO.9190 

55.  Internal  Contacts. — These  are  alluded  to  by  CHAUVENET  on 
pages  468  and  470,  but  are  not  clearly  explained.     As  these  and  the 


55  EXTREME   TIMES  GENERALLY.  71 

points  of  beginning  and  ending  are  special  cases  of  the  Rising  and 
Setting  Curve,  we  will  borrow  an  equation  from  the  next  section  to 
explain  them. 


4mp 

In  this  equation  all  the  quantities  are  positive  ;  p  is  the  radius  of 
the  earth,  /  that  of  the  shadow,  and  m,  as  we  will  see  presently,  is 
the  distance  between  the  two  cen- 
tres ;  p  and  /  are  sensibly  constant,  FIG.  5. 

so  that  m  is  the  only  variable.  As 
the  shadow  approaches  the  earth, 
m  is  very  large,  as  in  the  first  posi- 
tion, A,  in  Fig.  5  annexed  ;  hence, 
the  second  factor  of  this  equation  is 
negative,  and  the  value  of  X  imag- 
inary. It  is  not  until  this  factor 
becomes  o,  position  By  that  A  has  a 

real  value,  then  =  0,  which  is  the  beginning  or  end  of  an  eclipse,  for 
which  the  condition  is 

m  =  p  +  /.  (72) 

As  the  shadow  advances  it  crosses  the  primitive  circle  in  the  points 
a  and  6,  and  ^  is  the  angle  which  these  lines  aZ  and  bZ  make  with 
the  line  cZy  position  C.  If  internal  contacts  exist,  the  shadow  will 
become  tangent  to  the  primitive  circle  on  the  inside,  position  D,  and 
the  first  factor  will  become  0.  The  condition  for  these  is  then 

m  =  p  —  L  (73) 

As  the  shadow  advances  further,  m  becomes  smaller,  and  I  -f-  m  is 
smaller  than  p  numerically,  and  this  factor  then  becomes  negative 
and  X  again  imaginary,  position  E]  and  remains  so  until  m,  having 
passed  the  centre  of  the  sphere,  begins  to  increase,  when  I  -f-  m  =  +  P, 
and  we  have  the  second  internal  contact,  which  again  gives  equation 
(73),  and  the  above  succession  of  phenomena  is  reversed.  It  is  a 
beautiful  equation. 

The  formulae  for  the  internal  contacts  differ  from  those  for  exter- 
nal only  in  the  substitution  of  p  —  I  for  p  -f  I  in  equations  (56)  and 
(57).  They  are  not  computed  for  the  Nautical  Almanac  nor  shown 
on  the  charts  ;  we  will,  however,  recur  to  them  again  before  closing 
this  section. 

The  internal  contacts  exist  in  the  eclipse  of  1904,  Sept.  9,  occur- 


72  THEORY  OF   ECLIPSES.  55 

ring  at  about  7*  58m  and  9*  29ro,  as  shown  by  scale  in  Fig.  2,  Plate  I. 
They  do  not  exist  in  the  eclipse  of  1860,  July  19  (Fig.  3,  Plate  II.). 

56.  The  reader  must  not  confuse  these  four  contacts  with  the  four 
contacts  mentioned  by  CHAUVENET,  vol.  i.,  p.  440,  Art.  288,  which 
are  the  contacts  of  the  umbral  and  penumbral  cones  with  the  earth, 
and  near  the  points  K  and  L  (Figs.  2  and  3).    The  first  and  last  are 
the  same  as  we  have  just  considered. 

57.  Check  on  the   Extreme   Times. — A  rigorous   check  upon  the 
extreme  times  is  furnished  by  the  equation  (72),  computing  m  for 
the  beginning  and  ending  and  comparing  with  p  -j-  L 

Place 

m  sin  M  =  x  ") 
m  cos  M  =  y  ) 
Then  check 

(75) 

(76) 

Compute  with  five-place  logarithms  x  and  y  for  each  of  the  times 
T,  using  four  decimals  of  a  minute ;  then  compare  M  with  p,  which 
is  used  for  the  latitudes  and  longitudes  in  the  computation  above ; 
and  compare  log  m  with  log  (p  -f  /),  which  is  also  there  given.  The 
former  should  agree  within  3"  or  4",  and  the  latter  within  1  or  2 
units  of  the  last  decimal  place.  If  the  checks  agree,  it  shows  that 
these  quantities  and  also  the  times  are  correct. 

In  the  example  following,  the  factor  of  T  is  a  fraction  of  ten 
minutes. 

CHECK  ON  EXTREME  TIMES  GENERALLY. 

Beginning.  Ending. 

Nos.         log.  Nos.        log. 

(74)  T,  factor,      6*  7.8270    9.89360  11*  20.9190   8.96332 
Ax           -f  0.09259    8.96825  +  0.09292   8.96811 

+  0.07275   8.86185     -f-  .00854   7.93143 
x  tables         -  1.57629  +  1.39826 

x  -  1.50354  —  0.17712     -f  1.40680  +  0.14823 

Ay  —  0.02880  8.45939      -  0.02884   8.46000 

—  0.02254  8.35299      -  0.00265    7.42332 

y  table  +  0.31835             -  0.60406 

y  -f-  0.29581  +  9.47101     -  0.60671  —  9.78298 
tan  If 

(75)  M  281  7  48  0.70611     113  19  45   0.36525 
check  y  48                   46 

sin  N  9.99176  9.96296 

logm  0.18536  0.18527 

(76)  Check  log  p  -f  I  0.18536  0.18527 


57  EXTREME   TIMES  GENERALLY.  73 

The  extreme  times  may  also  be  found  in  a  very  simple  manner 
from  equation  72.     Place  it  under  the  following  form  : 

J  =  m  -  (p  +  0- 

Here  p  is  constant  and  I  nearly  so,  and  m  the  only  variable  ;  the 
extreme  times  occur  when  J  =  0.  We  can  get  p,  the  radius  of  the 
earth,  sufficiently  exact  from  Fig.  2,  the  angle  RZA  by  protractor 
measures  11°,  which  is  nearly  the  latitude  of  the  place,  since  the 
pole  is  but  slightly  elevated.  Table  IV.  gives  the  earth's  radius, 
which  in  numbers  is  0.99988  ;  I  from  the  eclipse  tables  is  0.53248, 
and  p  +  I  is  1.53236.  Next  find  x  and  y  for  every  ten  minutes  ;  the 
differences  will  have  to  be  extended  back  from  Qh  for  the  first  date. 
With  these,  using  five-place  logarithms,  compute  M  and  log  m  and 
get  m  in  numbers. 

Tabulating  these,  we  have  as  follows  : 

Time.          m.          m  —  (p  +  l).  M. 

5  50    1.70498  +  0.17262        QAQA  281°  44'  54"       1Q/    ,Q// 

6  0    1.60812  -f  0.07576  ~~  ™  -f-    7         281     25      5    '"  JX    S    —  2'  33" 
6  10    1.51133  —  0.02103  ~    ™'*        11         281       2    43          ^     ^     —  3      4 

6  20    1.41465  —  0.11771  ~  280    37    17 

By  formula  11,  Art.  21,  find  when  this  is  zero,  using  five-place 
logarithms.     In  numbers  the  terms  are 


t  =  _  -°75760  _  =  _  0.78265, 

—  .096790  —  .000045  +  .000026       .09681 

which  is  a  fraction  of  ten  minutes. 

Hence,  the  time  is  6*  7m.8265, 

And  from  the  regular  computation  we  get     6    7  .8270, 

which  is  sufficiently  exact  to  give  the  longitude  correctly.  With  this 
time,  interpolating  M,  it  is  found  to  equal  f  exactly  ;  hence,  the  lati- 
tudes and  longitudes  can  also  be  found. 

58.  Geometrical  Illustration.  —  We  already  have  the  sphere  and 
path  of  the  eclipse  projected  (Fig.  2,  Plate  I.,  and  Fig.  3,'  Plate  II.). 
With  a  pair  of  dividers  opened  to  the  radius  of  the  penumbra  /, 
place  one  leg  on  the  path  and  move  it  until  the  other  leg  sweeps  a 
circle  tangent  to  the  sphere  at  R  and  8.  The  point  a  on  the  path 
then  gives  the  time  of  beginning  generally,  and  similarly  b  that  of 
ending,  and  R  and  8  are  the  points  on  the  earth  of  first  and  last 
contact.  The  internal  contacts,  which  exist  only  in  Fig.  2,  are  found 
in  a  similar  manner  with  contacts  on  the  inner  side  of  the  primitive 


74  THEORY  OF   ECLIPSES.  58 

circle ;  they  are  marked  c  d  on  the  path.  The  path  may  now  be 
divided  into  10m  spaces,  or  closer  if  the  scale  of  the  drawing  admits, 
and  the  times  read  off;  the  points  of  contact,  R  and  S9  on  the  earth's 
surface,  are  those  given  in  the  Almanac  for  First  and  Last  Contact; 
their  latitudes  and  longitudes  can  easily  be  gotten  from  these  figures. 
If  we  mark  9*  (Fig.  2)  and  2h  (Fig.  3)  as  the  epoch  hour  T0,  and 
take  the  coordinates  of  T0,  the  epoch  hour,  as  XQ  and  T/O,  we  have 
(formula  54)  YZTQ  =  MQJ  measured  always  from  the  axis  of  Y  to  ward 
the  right  as  positive  ==  +  154°  10'  in  Fig.  2,  and  —  7°  46'  in  Fig.  3, 
and  TqZ=  m0.  Likewise  the  hourly  motions  give  the  angle  which 
the  path  makes  with  the  axis  of  Y.  This  is  seen  from  the  triangles 
having  H  at  the  right  angle,  but  as  y  is  negative  in  both  figures,  the 
angle  in  each  case  is  greater  than  90°  ;  and  if  J  be  the  point  where 
the  path  crosses  the  axis  of  Y,  CJb  =  N,  and  n  is  the  hypothenuse 
of  those  right-angled  triangles  (marked  H  at  the  right  angle) — the 
motion  of  the  shadow  along  the  path  in  one  hour.  N  is  also  meas- 
ured from  the  axis  of  Y  toward  the  right  as  positive,  and  is  never 
negative. 

In  Plates  I.  and  II. ,  as  well  as  in  subsequent  ones,  the  geomet- 
rical quantities  used  in  the  formulae,  lines,  and  angles,  etc.,  are 
marked  in  the  figures  with  thin  skeleton  letters,  which  should  not 
be  mistaken  for  the  heavier  reference  letters.  To  have  the  lines 
and  angles  marked  will  generally  render  the  description  clearer. 

The  reader  should  follow  the  formulae  now  as  closely  as  in  the 
example.  M—  N  is  the  angle  between  the  path  and  TQZy  measured 
from  the  path  toward  the  right  as  positive.  Draw  ZI  a  perpendicu- 
lar to  the  path,  forming  TQIZ,  a  right-angled  triangle  in  which  the 
angle  at  T0  ==  (M—  N\  and  T0Z  =  m0 ;  hence  ZI=  m0  sin  (M—  N)-, 
ZR  and  ZS  are  the  earth's  radii  =  p  ;  aR  and  bS  are  the  radii  of 
the  penumbral  shadow  /.  Hence  we  have  two  other  right-angled 
triangles,  aZI  and  bZI9  in  which  aZ  and  bZ  =  p  +  /,  and  Zl  =  mQ 
sin  (M —  N)  in  each,  from  which  we  have  the  angles  at  a  and  b 
given  by  their  sine  : 

.        =  IZ  =  m0  sin  (M-  N)  =  IZ 
~~aZ~  p  +  l  bZ 

And  here  we  see  the  ambiguity  as  to  which  angle  to  take — the 
acute  or  obtuse  value.  I  have  here  placed  these  angles  equal  to  one 
another,  which  is  sufficiently  accurate  in  the  drawing  and,  moreover, 
renders  the  explanation  more  easy.  Then  we  have 

0?  ~f~  0  cos  ^  =  a/=  Ib. 


58  EXTREME   TIMES  GENERALLY.  75 

This  is  reduced  from  space  to  time  by  dividing  it  by  n,  the  motion 
of  the  shadow  in  one  hour  along  the  path.  We  also  have  TQI  '  = 
m0  cos  (M  —  N\  and  this  is  also  reduced  to  time  by  dividing  by  n. 
Now  we  know  the  time  the  shadow  takes  in  passing  from  a  to  I 
and  from  /  to  b,  also  from  TQ  to  /.  Hence  the  time  numerically, 
and  r0  =  —  al—  T0I=  +  Ib  —  T0I.  We  know  the  time  at  T0,  the 
epoch  hour,  so  we  have  the  times,  T=  TQ  -j-  r. 

59.  For  the  middle  time,  it  is  readily  seen  why  it  is  given  by  the 
second  term  of  Equation  57. 


n 


-^)=  TJ, 


which  reduces  the  time  from  T0  to  the  point  J,  the  greatest  eclipse, 
which  is  also  the  Middle. 

60.  The  latitudes  and  longitudes  of  the  points  of  beginning  and 
ending  are  next  required.  The  formula  65  can  be  shown  geomet- 
rically ;  but  there  is  a  simpler  method  which  can  be  used  here,  since 
all  the  points  lie  in  the  fundamental  plane.  It  will  be  noticed, 
however,  that  f  =  N-\-  $  =  CZR  for  beginning,  which  is  always 
negative  if  taken  less  than  180°  ;  and  CZ8  for  ending,  which  is 
always  positive  ;  7-  throughout  the  eclipse  formulae  is  measured  from 
the  northern  part  of  the  axis  of  Y  toward  the  right  as  positive. 

The  method  of  getting  the  latitude  and  longitude  will  be  illus- 
trated by  only  one  point  —  that  for  ending,  which  is  located  in  Fig.  2 
by  the  letter  8.  Revolve  the  whole  sphere  90°  around  the  line  CD 
as  an  axis  ;  the  pole  P  will  fall  at  P',  as  in  Art.  39.  P'Z  produced  is 
the  earth's  axis.  8  will  fall  at  m  in  the  line  CD  on  the  lower  sur- 
face of  the  sphere,  and  E  at  E1.  As  the  pole  is  in  the  principal 
plane,  all  the  circles  of  latitude  will  project  in  right  lines  perpen- 
dicular to  the  axis  P'Z,  hence  E'E"  will  be  the  equator,  and  mw, 
crossing  the  axis  at  £,  and  at  right  angles  to  it,  will  be  the  parallel 
of  latitude  of  the  point  8,  and  the  latitude  can  be  measured  by  a 
protractor,  the  arc  E"n. 

For  the  longitude,  resume  the  sphere  in  its  normal  position  and 
revolve  it  around  the  line  AB  until  the  pole  falls  in  the  point  Z; 
the  parallels  of  latitude  will  then  project  in  circles.  We  have 
the  radius  of  the  parallel  of  the  point  8,  which  is  in;  with  this 
describe  an  arc,  o,  from  Z  as  a  centre,  and  it  will  be  the  required 
parallel.  During  this  revolution  the  point  8  will  revolve  in  the 
vertical  circle,  Sq,  below  the  primitive  plane,  and  fall  at  s,  where 


76  THEORY   OF   ECLIPSES.  60 

this  line  meets  the  circle  of  latitude.  In  the  present  case  the  line 
Sq  meets  the  circle  at  such  an  acute  angle  that  the  point  s  is  rather 
indefinite ;  and  another  construction  is  necessary  to  determine  how 
far  S  has  revolved.  With  the  sphere  in  its  normal  position,  inter- 
sect it  by  a  plane  through  Sq,  which  will  cut  a  small  circle  from  the 
sphere,  whose  diameter  is  Sq.  This  circle  revolve  down  so  that  the 
part  above  the  line  AB  falls  at  z.  We  can  now  see  the  revolution. 
The  pole  was  revolved  through  an  arc  90°  —  d  =  90°  —  5°  15'  = 
84°  45'.  S  at  the  same  time  revolved  through  the  same  arc  and  fell 
at  r.  Its  motion  along  the  line  Sq  is  the  sine  of  this  angle  found 
by  projecting  r  on  the  line  at  s.  Therefore,  we  have  the  hour 
angle,  DZs,  measured  in  a  positive  direction.  The  Greenwich 
Meridian  at  the  time  of  ending  is  on  the  circle  of  the  sphere  about 
one-third  distant  from  11*  to  12*,  and  //x  is  the  angle  from  this  point 
to  D  ;  and  the  longitude,  the  arc  from  the  position  of  Greenwich  to 
the  line  ZS  extended. 

The  drawing  is  too  crowded  for  more  lines ;  but  the  results  for 
the  inner  contacts,  taken  from  the  figure  by  scale  and  protractor, 
are  as  follows : 

T  2d  Contact  7*  58m.5  3d  Contact  9*  28m.7 

<P  —  3°  30  —  32°  30 

#  —  89  30  +85  30 

/*!  +120  20  142  50 

+  20950 

-150  10 

They  plot  well  on  the  chart  in  the  Nautical  Almanac,  which  is 
reproduced  in  Fig.  17,  Plate  VI. 

Differences  with  the 
Middle. 
1    36.6 
0    45.9 

0  44.3 

1  36.5 


Comparison  of 
the  Times  of  - 
the  Contact. 

'  First  Contact        6*    7m.S 
Second  Contact    7   58  .5 
Middle                  8   44  .4 
Third  Contact      9   28  .7 
Last  CoDtact      11   20  .9 

61  RISING  AND  SETTING  CURVE.  77 

SECTION   VI. 
KISING  AND  SETTING  CURVE. 

61.  CHAUVENET'S  formulae  are  as  follows  : 

m  sin  M  =  x 

m  cos  M  =  y  (77) 


sin  U  =  dt  -.-  (78) 

4pm 


(79) 
tan  r'  =  Pi  tan  r  (80) 

(81) 
sm  Y          cos  Y 

cos  . .   " 

(82) 

tan  f  =  ^EE£r  (83) 

-  =  ft-»  (84) 

m  sin  (3f —  E)  <p  sin  (y  —  E),  Eclipse  beginning,  ^  ,nf-\ 


sin  (M  —  E)  >  p  sin  (j  —  jE),  Eclipse  ending.       j 

#  between  0  and  —  180°,  The  sun  rising, 
&  between  0  and  +  180°,  The  sun  setting 


between  0  and  —  180°,  The  sun  rising,   |  xggx 

insr.  j 


CHATJVENET  intends  these  formulae  to  be  used  rigorously  with 
five-place  logarithms,  to  seconds  of  arc,  making  two  approxima- 
tions ;  taking  in  the  first  p  =  unity,  and  repeating  the  work  for  ^ 
and  f  •  then  proceeding  to  the  latitudes  and  longitudes. 

62.  There  is,  however,  no  necessity  for  this  exactness,  as  the 
curve  is  generally  only  needed  for  plotting  on  a  chart,  and  the  fol- 
lowing simplifications  may  be  made  :  Neglect  the  compression  of  the 
earth,  then  e  =  0,  and  the  following  changes  result :  p  —  1,  pl  =  1, 
r'  =  T,  <pi  =  <p,  and  take  d  instead  of  d^  This  will  give  results 
almost  or  quite  as  close,  as  can  be  seen  on  a  projection  of  the  sphere 
of  sixteen  inches  radius,  or  thirty-two  inches  in  diameter.  Moreover, 


78  THEORY   OF   ECLIPSES.  62 

equation  (85)  need  never  be  computed,  since  both  (85)  and  (86)  are 
known  when  the  curves  are  plotted  on  a  chart.  And  as  d  never 
varies  more  than  3'  or  V  of  arc,  it  may  be  taken  as  constant,  using 
the  value  at  the  epoch  hour. 

For  these  approximations,  four-place  logarithms  to  minutes  will 
be  sufficient,  and  compute  for  every  ten  minutes  between  the  extreme 
times.  The  double  sign  of  /  will  give  two  points  to  the  curve ;  the 
computation  can  be  made,  using  one  value  throughout  all  the  10-min- 
ute  columns,  finding  <p  and  to  ;  then  computing  with  the  other  value. 
(For  the  derivation  of  this  equation  (78),  see  Art.  89.)  No  example 
is  given  of  the  use  of  these  formulae,  as  they  are  so  similar  to  those 
of  the  preceding  section. 

If  a  pair  of  dividers  be  opened  to  the  radius  I  and  one  leg  moved 
upon  the  path  to  the  successive  10-minute  points,  the  other  leg  will 
generally  be  near  enough  to  touch  the  primitive  circle  from  each  of 
these  10-minute  points.  These  points  on  the  circle  are  the  two  posi- 
tions given  by  the  formulae  for  each  of  the  assumed  times  on  the 
path.  In  Fig.  3,  Plate  II.,  the  curve  is  seen  to  be  continuous,  the 
centre  line  being  distant  from  the  primitive  circle  at  no  part  of  its 
path  as  much  as  /;  but  in  Fig.  2,  Plate  L,  the  path  lies  so  near  the 
centre  of  the  earth  that  it  is  at  some  of  its  parts  at  a  greater  distance 
than  I  from  the  primitive  circle ;  consequently  the  dividers  cannot 
reach  this  circle,  and  there  is  no  rising  and  setting  curve  to  these 
times  of  the  path.  By  moving  the  dividers  successively  over  the 
10-minute  points  and  ascertaining  which  point  on  the  primitive  circle 
the  other  leg  reaches,  the  reader  will  see  how  the  inner  constants  are 
formed  in  Fig.  2,  and  the  reason  also  why  they  do  not  exist  in  Fig.  3. 

63.  The  above  formulae  may  be  partially  illustrated  as  follows  :  In 
Fig.  6— 

The  coordinates  of  centre  of  shadow  are  m  sin  M=  x 

m  cos  M=y 

Coordinates  of  point  of  the  curve  on  the  earth,    p  sin  y  =  £ 

p  cos  Y  =  "n 

Fundamental  equations  of  eclipses  (CHAUVE-    I  sin  Q  =  x  —  £ 
NET,  i.,  447,  449),  I  cos  Q  —  y  —  7j 

A  is  here  equal  to  /. 

Q  is  the  angle  of  position  of  the  centre  of  the  shadow  from  any 
point  on  the  cone  of  shadow,  and  is  measured  from  the  axis  of  Y 
toward  the  right  as  positive.  The  meanings  of  the  quantities  are 


63 


RISING   AND   SETTING   CURVE. 


Fja  6- 


clear  from  the  figure,  and  the  transformations  to  equation  (78)  for 
J  X  is  merely  to  get  this  angle,  a  con- 
venient form   for  use.     (See  Art.  64 
following.) 

As  the  radius  of  the  shadow  is  but 
little  more  than  half  the  earth's  radius, 
A  can  never  much  exceed  30°  or  35°. 

64.  Dr.  Hill9 a  Formulce.—The  fol- 
lowing formulae  were  devised  by  Dr. 
George  W.  Hill  for  this  curve.  Their 
derivation  is  given  in  Section  "VIII., 
as  they  result  directly  from  formulae 
for  the  outline  curves  : 

m  sin  M  =  x  ) 
m  cos  M  =  y  ) 

m2+l 


cos  (0  —  M}  = 


2m 


cos  <p  sin  $  =  sin  6 
cos  <p  cos  #  =  —  sin  d  cos  6 
sin  <f>  =  cos  d  cos  0 


(87) 
(88) 

(89) 
(90) 


Use  four-place  logarithms  to  minutes,  and  compute  for  every  ten 
minutes  between  the  extreme  times.  As  x  and  y  are  required  to  five 
places  of  logarithms  in  the  central  curve,  it  is  well  to  get  them  here 
to  five  places  for  the  times  between  the  limits  when  the  centre  of  the 
shadow  crosses  the  primitive  circle  ;  then  tan  M  need  be  taken  only 
to  four  decimals.  I  — I2  is  a  constant  for  the  curve;  (d  —  M )  has 
two  values,  -j-  and  — ,  as  in  CHAUVENET'S  formula,  which  gives  two- 
points.  As  d  varies  only  a  few. minutes  during  the  whole  eclipse, 
sin  d  and  cos  d  may  be  taken  as  constant  near  the  middle  time. 
Finally,  instead  of  getting  tan  <p,  take  the  angle  from  the  sine  and 
notice  whether  the  cosine  gives  the  same  angle.  Their  agreement 
will  check  several  errors,  and  the  computer  may  remember  that  two 
points  separated  by  10'  or  15'  of  arc  can  hardly  be  seen  on  a  chart 
of  which  the  radius  of  the  sphere  is  sixteen  inches.  The  heavy  lines 
on  the  eclipse  drawings  prepared  for  the  Almanac  are  about  15'  wide. 

65.  Example.— One  point  of  the  eclipse  of  1904,  Sept.  9,  at  7h  10% 
is  taken,  giving  two  geographical  positions.  Each  column  should  be 


80 


THEORY   OF   ECLIPSES. 


65 


headed  with  the  hour  and  10m.  The  example  here  given  should  be 
carried  out  in  one  column.  The  constants,  which  are  used  on  a  slip 
of  paper,  are  here  written  in  the  margin  of  the  work.  Repetition 
of  figures  throughout  the  whole  eclipse  computation  is  avoided  as 


much  as  possible.     With  log  J  =  9.6990,  log 


is  gotten  as  a 


constant  and  added  to  the  quantity  B  of  the  addition  and  subtraction 
logarithms.  (6  —  M  )  has  the  double  sign,  which  gives  two  values  to 
6  and  thence  two  geographical  positions  for  the  two  branches  of  the 
curve  at  this  time. 


EXAMPLE,  RISING  AND  SETTING  CURVE. 


(87 )  Numbers       —  0.9256        log  x 
4-  0.1167 


(89) 


log  sin  d         4-  8.9612 


log  cos  d 
(90)  ft 


9.9982 
108°  IV 


logx 

—  9.9664 

logy 

4-  9.0671 

tanM 

0.8993 

M 

—  82   49 

sin  M 

9.9966 

logm 

4-  9.9698 

logm2 

4-  9.9396 

I  —  I 

A  0.0844 

B  0.3453 

log 

4-  9.8995 

cos(0  —  3f) 

9.9297 

(6-M) 

=F  31°  44' 

e 

—  114   33 

—  51     5 

sin 

—  9.9588 

—  9.8910 

cos 

—  9.6186 

4-  9.7981 

COS  0  COS  # 

4-  8.5798 

—  8.7593 

tan# 

1.3790 

1.1317 

# 

—  87    36 

—  94   14 

sin 

9.9996 

9.9988 

COS0 

4-  9.9592 

4-  9.8922 

sin  <j> 

—  9.6168 

4-  9.7963 

<j> 

—  24°  27' 

4-  38°  44' 

u 

4-  195   47 

4-  202   25 

66.  The  Node  or  Multiple  Point. — This  is  the  point  where  the  two 
branches  of  the  rising  and  setting  curve  cross  oue  another.  It  is 
seen  in  Fig.  3,  but  not  in  Fig.  2,  as  it  generally  does  not  exist  when 
the  inner  contacts  are  formed.  The  points  of  the  two  curves  are 
passed  over  by  the  cone  of  shadow  at  widely  different  times,  and  the 
method  of  its  formation  is  given  in  the  next  section  in  connection 
with  the  Maximum  Curve.  It  is  an  unimportant  point,  but  should 
be  properly  located  on  the  chart.  It  is  very  near  the  meridian  on 


66  KISING  AND   SETTING   CURVE.  81 

which  the  sun  is  central  at  local  apparent  noon,  so  we  may  take  with 
sufficient  exactness, 

<p  =  90°  -  d,  (91) 

o»  =  jii.  (92) 

If  the  shadow  goes  over  the  pole  instead  of  the  latter, 

w  =  ^  +  180°.  (93) 

67.  Singular  Forms  of  the  Rising  and  Setting  and  Other  Curves. — 
The  two  most  usual  forms  of  the  rising  and  setting  curve  are  shown 
in  Fig.  3,  the  distorted  figure  8,  and  in  Fig.  2  the  two  separate 
branches  connected  by  two  limiting  curves,  which  are  more  readily 
recognized  in  Fig.  17,  Plate  VI.,  and  Fig.  18,  Plate  VII.,  repro- 
duced from  the  eclipse  charts.  Other  special  forms  are  as  follows : 

A.  When  the  rising  and  setting  curve  breaks  to  form  two  limiting 
curves,  the  break  does  not  necessarily  take  place  at  the  node.     This 
is  well  illustrated  in  the  eclipse  of  1894,  Sept.  28  (Fig.  7,  Plate  IV.), 
at  A,  where  the  whole  shadow  very  nearly  fell  upon  the  earth ;  the 
rising  and  setting  curve  contracted  at  this  point,  then  diverged  before 
crossing  at  the  node.     If  the  shadow  had  fallen  a  very  little  further 
north,  the  break  would  have  occurred  at  the  point  a,  forming  three 
branches,  one  of  which  would  extend  from  some  point  near  a  to  the 
node.     The  form  of  the  curve  is  shown  in  the  figure. 

B.  partial  eclipse,  March  26,  1884.     Here  the  southern  limit  of 
shadow  passed  but  a  short  distance  to  the  right  of  the  axis  Y9  conse- 
quently that  branch  was  very  small,  as  shown  at  J9,  Fig.  7.    The  next 
eclipse  of  the  series,  1902,  April  8,  for  which  no  chart  was  given  in 
the  Almanac,  had  but  one  branch  of  the  rising  and  setting  curve 
shown  at  b. 

C.  Let  this  line  (Fig.  7)  represent  the  centre  line  of  the  annular 
eclipse  of  1891,  June  6  ;  the  pole  p  being  below  the  line,  the  shadow 
has  consequently  passed  over  and  beyond  the  pole,  and  the  "  Central 
Eclipse  at  Noon/'  as  given  in  the  Nautical  Almanac,  is  more  properly 
"  Central  Eclipse  of  the  Midnight  Sun."    In  this  eclipse  also  another 
peculiarity  is  noticed,  the  path  of  the  Annulus  passed  from  East  to 
West. 

Let  D  (Fig.  7)  represent  the  centre  line  of  the  annular  eclipse, 
1896,  Feb.  13.  It  is  seen  that  it  is  so  far  south  that  it  does  not  cross 
the  principal  meridian  ;  there  is  consequently  no  "  Eclipse  at  Noon," 
as  generally  given  in  the  Almanac. 

E.  This  line  (Fig.  7)  represents  the  path  of  the  Total-Annular 
Eclipse,  1890,  Dec.  11.  Here  the  vertex  of  the  umbral  cone  fell 

6 


82  THEORY   OF   ECLIPSES.  67 

just  above  the  fundamental  plane,  so  that  the  ends  of  the  eclipse 
were  annular  and  the  middle  total,  the  limiting  curves  crossing,  as 
shown  in  the  figure.  Professor  NEWCOMB  selected  the  name  Central 
for  this,  as  given  in  the  Almanac. 

F  (Fig.  7).  The  central  path  of  the  annular  eclipse,  1874,  Oct.  7, 
passed  over  so  small  a  portion  of  the  earth  that  the  line  of  central 
eclipse  lay  wholly  within  the  rising  and  setting  curve,  as  shown  at  F. 

G.  The  partial  eclipse  of  1862,  Nov.  21,  presents  a  curious  anom- 
aly :  the  southern  pole  of  the  earth,  pf,  was  elevated  almost  to  its 
extreme  limit,  the  shadow,  as  much  as  fell  on  the  earth,  was  wholly 
beyond  the  pole,  p'y  so  that  the  northern  limiting  curve  was  the 
nearest  point  of  this  eclipse  to  the  south  pole,  and  all  of  the  shadow 
passed  over  the  earth  from  east  to  west.  No  chart  is  given  in  the 
Almanac,  but  sufficient  latitudes  and  longitudes  to  construct  the 
eclipse  as  here  shown. 

H.  The  limiting  lines  of  the  central  eclipse  sometimes  show  a 
peculiarity ;  the  angle  E  at  the  ends  skews  the  points  of  this  curve 
so  much  at  the  ends  that  the  southern  point,  A,  may  have  a 
greater  latitude  than  the  corresponding  northern  point,  as  the  case 
of  the  total  eclipse  of  1896,  Aug.  8,  which  is  the  same  eclipse  as 
shown  in  the  chart  (Plate  VII.,  Fig.  18).  See  also  Art.  139. 


SECTION    VII. 

MAXIMUM  CURVE. 

68.  WITHIN  the  space  formed  by  the  two  branches  of  the  rising 
and  setting  curve,  on  which  the  eclipse  commences  and  ends  in  the 
horizon,  there  must  be  a  succession  of  places  where  the  eclipse  is  seen 
at  its  maximum,  which  forms  the  curve  known  as  the  Maximum  of  the 
Eclipse  in  the  Horizon.  CHAUVENET'S  formulae  are  as  follows  : 

m  sin  M  =  17  ) 
m  cos  M  =  y  ) 

m  sin  (M—  E)  ,ft_. 

sm  ^  = * (95) 

P 
Take  here 

log  p  =  i  log  p!  (96) 


68  MAXIMUM   CURVE.  83 

J  to  be  positive, 

±  A  =  m  cos  (M—  E)  —  p  cos  <I>  (97) 

*<l  (98) 

r=^  +  E  (99) 

tan  Y'  =  Pi  tan  ^  (100) 

sin  Yf      p\  cos  Yr 

P  =  -H-  =  —  (ioi) 

sin  f         cos  Y 
Then  repeat  from  Equation  (95)  for  a  correct  value  of  -f 

cos  ft  sin  &  —  sin  y'  ^v 

sin  ^i  cos  *  =  —  cos  Y'  sin  e^  >  (102) 

sin  f>!  =  cos  Yr  cos  c?!      3 

tan  ft 
tan  o>  = _ 


«  =  /«i  —  *  (104) 

As  in  the  preceding  section,  there  is  no  need  of  such  rigor  in 
these  equations ;  CHAUVENET'S  method  is,  however,  given  in  full. 
Two  approximations  are  necessary,  getting  p  correctly  from  the 
first ;  then  repeating  equations  (95),  (99),  (100),  and  (101),  and  the 
remainder  for  the  latitudes  and  longitudes.  Four-place  logarithms 
are  sufficient. 

69.  Remarks  on  Formulas,  and  Chauvenet's  Text. — Some  of  these 
formulae  may  need  explanation  to  the  beginner,  and  one  paragraph 
of  CHAUVENET'S  text  needs  revision,  as  it  gives  a  wrong  explana- 
tion of  the  beginning  and  ending  of  this  curve. 

The  equation  log  p  =  %  log  pl  is  a  mean  value  derived  thus  :  On 
page  477  of  CHAUVENET  we  have  these  equations  : 


,,=i     \ 

Pl       C 
e»4V-iJ 


(105) 
p  cos  Y  =  7  = 


Adding  the  squares 


'  sin2  Y  +  —*  cos2  Y  =  ?  +  if  =  1 
Pi 


84  THEORY  OF  ECLIPSES.  69 

The  extreme  values  of  7-  are  0  and  90°. 

If  f  =  0        p  =  ft  If  r  =  90°        p=l 

The  mean  of  these  is 

Pi  -f  1  ft3  +  2ft  +  1 

|j  =  r1        »  whence  ^a  =  r          r    —  =  ft 


In  the  latter  equation  pl  is  larger  than  its  square  and  smaller  than 
unity,  therefore  the  numerator  is  approximately  4plt 
Passing  to  logarithms 

2  log  p  =  log  ft,  whence        log  p  =  £  log  ft. 

On  page  476  of  CHAUVENET,  lines  10  and  11,  there  is  an  error 
of  signs  —  those  between  the  terms  should  be  transposed  :  In  the 
first  equation  the  sign  should  be  +,  and  in  the  second,  —  . 

In  Art.  309,  page  478,  is  the  following  paragraph  :  "  The  lim- 
iting times  between  which  the  solution  is  possible  will  be  known 
from  the  computation  of  the  rising  and  setting  limits,  in  which  we 
have  already  employed  the  quantity  m  sin  (M  —  E)  ;  and  the  pres- 
ent curve  will  be  computed  only  for  those  times  for  which  m  sin 
(M  —  E)<1.  These  limiting  times  are  also  the  same  as  those  for 
the  northern  and  southern  limiting  curves,  which  will  be  deter- 
mined in  Art,  313." 

The  paragraph  is  wrong  throughout.  The  limits  of  the  rising 
and  setting  are  really  the  extreme  times  ;  but  by  these  words  CHAU- 
VENET means  the  extreme  times  of  the  northern  and  southern  lim- 
iting curves  ;  and  these  are  not  the  limiting  times  of  the  maximum 
in  all  cases.  They  are  in  CHAUVENET'S  example,  but  not  in  the 
eclipse  of  1904,  September  9.  The  equation  m  sin  (M  —  E)  <? 
should  read  <  unity.  Compare  page  479,  line  2. 

70.  The  Maximum  Curve  is  correctly  explained  thus  :  As  the 
earth  is  moving  under  the  shadow  in  some  different  direction,  the 
projection  of  the  path  over  the  earth's  surface  will  not  be  its  pro- 
jection on  the  fundamental  plane,  but  will  be,  perhaps,  the  result- 
ant of  the  two  motions.  The  northern  and  southern  limiting  curves 
will  consequently  not  be  formed  by  the  element  of  the  cone  normal 
to  the  path,  but  by  some  other  element  swung  around  from  the  right 
angle  by  the  angle  E  +  some  variable  quantity.  At  the  beginning 


70  MAXIMUM   CURVE.  85 

and  ending  of  these  curves  this  variable  is  zero,  and  Q  =  E  or  Q  = 
E  ±180°  (CHAUVENET,  i.,  p.  481,  Art.  311).  Hence,  for  the  be- 
ginning of  the  Maximum  Curve,  conceive  a  line  crossing  the  path  of 
the  eclipse,  making  the  variable  angle  E  with  the  axis  Y.  As  this 
line  moves  along  the  centre  line,  its  intersection  with  the  earth's 
sphere  will  generate  the  Maximum  Curve  in  all  cases ;  and  there 
are  four  special  positions  to  be  examined. 

In  Fig.  8,  Plate  V.,  which  is  a  reproduction  of  Fig.  2,  the  eclipse 
of  1904,  September  9,  draw  the  line  ZW,  making  the  angle  E  =  CZW 
with  the  axis.  Also  draw  ab  from  the  path  tangent  to  the  sphere 
at  6  and  parallel  to  ZW,  also  making  the  angle  E  with  the  axis. 
Then  in  Equations  (94)  and  (95)  we  have,  angles  measured  from  the 
axis  toward  the  left  being  negative  : 

CZa  =  M,  which  is  negative  in  this  case. 
CZW=  E,  which  is  positive. 
WZa  =  M  —  E,  which  is  negative  and  obtuse. 
Za=z  m. 

ac  =  ra  sin  (M  —  E\  negative, 
=  Zb,  since  ab  is  parallel  to  c  W 
=  />,  the  radius  of  the  earth's  sphere. 

„           .            ra  sin  (M — E)       p  1,1  .•     i    *        v 

Hence  sin  <f>  = — -  =  -  =  —  1,  taken  negatively  from  Z 

P  P 

for  beginning. 

=  WZb  =  90°  in  this  position. 


Here  we  see  the  errata  made  in  CHATJVENET'S  clause  above 
quoted;  as  p==  py  the  earth's  radius,  m  sin  (M —  E)  equals  unity, 
very  nearly. 

Just  before  the  line  ab  became  tangent  to  the  sphere,  m  had  a 
larger  value,  sin  <j>  >  unity,  and  <p  therefore  imaginary,  giving  no 
points  to  the  curve.  This  is  the  first  position  above  referred  to.  The 
maximum  curve  therefore  begins  at  b  when  sin  <p  =  unity.  Imme- 
diately after  the  first  contact,  as  the  line  ab  advances,  it  cuts  the 
sphere  in  two  places,  and  there  are  double  points  to  the  maximum 
curve,  which  CHATJVENET  refers  to,  but  does  not  clearly  explain. 
These  double  points  exist  until  the  line  ab  takes  the  position  de,  in 
this  example,  in  which  e  is  the  point  of  beginning  of  the  northern 
limiting  curve,  which  CHAUVENET  vaguely  calls  the  "limiting 
times  "  of  the  maximum  curve.  A  double  point  at  7h  I0m  is  given 
in  the  example  below. 


86  THEORY   OF   ECLIPSES.  71 

71.  To  digress  for  a  moment,  it  will  be  noticed  that  these  points, 
marked  VW  in  Fig.  3,  are  the  extreme  points  of  the  southern  limit- 
ing curve  ;  and  in  this  figure  are  the  "  limiting  times  "  of  the  maxi- 
mum, as  CHAUVENET  writes  while  having  this  eclipse  in  his  mind, 
but  the  statement  is  not  general  as  applying  to  all  eclipses. 

72.  Take  a  second  position,  the  line  de  through  the  extreme  point 
e  of  the  northern  limiting  curve.     The  angles  M  and  E  are  hardly 
changed,  since  the  former  position 

WZd  —  M  —  Et  negative  and  obtuse. 
Zd  =  m. 

dc=m  sin  (M  —  2£),  negative, 
=  d'Z. 

d'Z      d'Z 


sm     = 


Ze         p 

This  equation  gives  two  values  of  ^,  according  as  the  cosine  is 
taken  positive  or  negative,  and  these  give  the  double  points  to  the 
curve  when  they  exist.  We  can  take  the  two  cases  together  ;  laying 
off  the  arc  bl  =  be,  we  have 

With  cos  0-f,  </>  =  WZe, 
With  cos  <}>  —  ,  <p  =  WZl. 

And  by  formulae  (97)  and  (98)  for  the  criterion, 

Zd  =  wi, 

Zc  =  m  cos  (M  —  E~)  and  negative. 

{Zf  =  p  cos  ^,  positive,  since  cos  (p  is  positive, 
Zk  ==•  p  cos  </>,  negative,  since  cos  0  is  negative. 
J  =  m  cos  (  M  —  E)  —  p  cos  <p. 

{=  Zc  +  Zf  =  cf,  the  numerical  sum, 
=  Zc  —  Zk  =  ck,  the  numerical  difference. 

In  the  first  case,  J  =  /,  as  it  should,  since  we  have  taken  the  point 
at  which  the  maximum  ends. 

In  the  second  case,  J<  I,  and  there  is  a  double  point  to  the  curve 
located  at  /. 

As  the  shadow  moves,  the  northern  branch  gives  J  >  I,  and  the 
condition  is  not  fulfilled  ;  but  there  are  single  points  to  the  southern 
branch,  and  this  is  the  third  position  referred  to.  Single  points  exist 
until  the  line  ab  reaches  the  position  gh,  after  which  for  both  branches 


72  MAXIMUM   CURVE.  87 

and  the  condition  is  not  fulfilled  by  either,  yet  the  formula  gives  real 
points,  though  they  lie  outside  of  the  limits  of  the  curve.  In  fact,  they 
extend  round  the  whole  circle  of  the  earth's  sphere.  This  is  the  fourth 
position,  and  to  exclude  these  values  the  condition  (98)  is  imposed. 

It  is  sometimes  an  advantage  to  compute  one  of  these  points  out- 
side of  the  eclipse ;  or  when  the  points  of  the  maximum  curve  lie 
far  apart,  and  if  there  is  no  point  near  the  end,  a  point  outside  may 
be  utilized  to  give  the  direction  of  the  line  on  the  chart,  or  to  check 
the  extreme  point. 

Generally  the  criterion  (97),  (98),  need  not  be  computed,  and  never 
except  toward  the  ends  of  the  curve.  If  a  plot  of  the  eclipse  similar 
to  those  here  given  is  made,  it  will  answer  every  purpose.  A  point 
outside  of  the  eclipse  may  perhaps  be  inadvertently  computed,  but 
an  examination  of  the  times  on  the  chart  will  show  if  it  belongs 
there.  The  lines  from  the  path  de,  gh,  etc.,  must  each  make  the 
correct  angle  of  the  variable  E  at  the  times  marked  by  d  and  g  on 
the  path. 

73.  Fig.  3,  which  is  CHAUVENET'S  eclipse,  shows  a  very  simple 
maximum  curve ;  the  lines  from  the  path  making  the  angle  E  are 
drawn,  which  gives  the  times  of  beginning  and  ending.      In  this 
eclipse  they  are  the  same  as  the  extreme  times  of  the  southern 
limiting  curve. 

74.  For  the  geographical  positions,  we  have  above  for  the  point  e 
the  angle  (p. 

Zed  =  <p  =  eZW,  negative  angle. 
Then   CZe  =  eZW  —  CZW  =  4>  —  E  =  r,  negative  angle. 

From  this  equation  f  is  found  for  use  in  getting  the  geographical 
positions,  which  are  completely  known  when  this  angle  is  determined 
and  the  time  known.  The  formulae  are  similar  to  those  in  previous 
sections. 

75.  Approximate  Formulae. — This  curve  should  be  placed  below 
the  rising  and  setting  curve.    Since  m  and  M  are  already  there  com- 
puted, we  then  proceed — 

sin  4>  =  m  sin  (M  —  E).  (106) 

r=<P  +  E.  (107) 
cos  <p  sin  #  =  sin  y,                ^ 

cos  <f>  cos  #  =  —  cos  Y  sin  d,  >  (108) 
sin  <f>  =  cos  Y  cos  d.       ) 

a,  =  Ml  —  *.  (109) 


88 


THEORY   OF   ECLIPSES. 


75 


As  in  the  former  curve,  neglecting  the  compression  of  the  earth, 
e  =  01?  and  there  results  p  =  1,  7*  =  7*',  <P  =  (Pi)  an(i  d  —  d\-  Jtf  and 
m  are  to  be  taken  from  the  rising  and  setting  curve.  The  criterion 
(97)  and  (98)  will  seldom  be  needed ;  it  is  here  given  in  the  example 
merely  to  show  the  double  point,  f  is  always  <C  90°  north  of  the 
point  Zj  and  >  90°  south  of  it.  If  taken  less  than  180°,  it  is  nega- 
tive on  the  left  of  the  axis  and  positive  on  the  right,  f  and  (p  will 
usually  lie  in  the  same  quadrant.  (p  here  has  no  affinity  to  the  angle 
(p  in  beginning  and  ending  generally. 


76.  EXAMPLE — MAXIMUM  CURVE. 
TOTAL  ECLIPSE  1901,  SEPTEMBER  9. 


E=  +15° 

7"     iu"1 

2'      M—E—W     51 

(107) 

V    —67°  32'—  112°  28/ 

(106) 

sin(     ) 

—9.9959 

(108) 

7 

—52    30 

—97      26 

cos  (     ) 

—9.1354 

sin 

—9.8995 

—9.9963 

sin  V 

—9.9657 

cos 

+9.7844 

—9.1118 

(D 

m  cos  (     ) 

—9.1052 

log  sin  d  +  8.9612  \ 

cos  <j>  sin  # 

—8.7456 

+8.0730 

(2)  cos  ^ 

+9.5822 

tan  # 

—1.1539 

—1.9233 

(97) 

I  0.5326 

Nos.  (1) 

—0.127 

# 

—94     1 

—89      19 

Nos.  (2) 

+0.383 

sin  cos 

9.9989 

0.0000 

(98) 

Double  point  shown 

cos  <;> 

+9.9006 

+9.9963 

Since  (1)  - 

H2)<* 

log  cos  d 

9.9982        sin  <j> 

+9.7826 

—9.1100 

And    (1)_(2)<J 

Pi 

108°  llr        4> 

+37    19 

—7     24 

w 

+202  12 

+197  30 

In  the  criterion  the  sign  of  m  cos  (M — E)  is  determined  term  (1), 
that  of  (p  undetermined,  except  by  the  criterion,  in  which,  as  these 
terms  are  to  be  subtracted,  they  will  generally  have  the  same  sign, 
so  as  to  make  their  difference  small  and  less  than  I.  This  will  de- 
termine the  sign  of  (p  generally  in  practice.  In  this  example  the 
second  value  of  <p,  that  in  the  last  column,  is  the  value  correspond- 
ing to  the  succession  of  points,  and  the  acute  value  the  special  case. 
The  criterion  determines  whether  the  angle  is  acute  or  obtuse,  but 
the  sign  in  either  case  is  determined  by  that  of  sin  (p  above  in  the 
formulae. 

77.  The  Node  and  Maximum  Curve. — The  Maximum  Curve  does 
not  pass  through  the  node,  though  in  all  the  earlier  charts  of  the 
Almanac  it  is  drawn  so. 

The  node  is  formed  in  the  following  manner  :  As  the  shadow  ad- 
vances, the  pole,  P,  being  elevated  as  in  the  examples  above  given, 


77  MAXIMUM   CURVE.  89 

the  preceding  arc  of  the  shadow  will  pass  over  some  point,  a,  on  the 
right  of  the  axis.  When  the  follow- 
ing arc  of  the  shadow  approaches,  ' 
the  point  a  has  reached  b  just  as 
the  arc  passes  over  it.  The  point 
a  in  this  case  revolves  below  the 
sphere,  since  the  pole  is  elevated. 
The  motions  of  the  shadow  and 
point  a  may  be  considered  as  con- 
stant ;  and  when  the  centre  of 
the  shadow  has  moved  half  of  its  distance  and  is  at  c,  the  point 
a  has  also  revolved  half  of  its  distance  and  is  on  the  meridian 
below  the  sphere,  the  maximum  curve  is  being  formed  in  the  funda- 
mental plane,  and  it  is  thus  seen  that  it  cannot  pass  through  this 
nodal  point.  When  the  centre  of  the  shadow  is  at  c?,  the  point  a 
has  moved  to  the  left  of  the  meridian.  This  is  now  the  meridian  of 
the  sun  at  noon,  and  the  point  a  is  about  180°  from  it. 

If  the  pole  is  depressed,  the  shortest  arc  for  the  point  a  to  move 
in  will  be  from  b  to  a,  passing  above  the  fundamental  plane,  and 
the  maximum  is  again  formed  in  the  fundamental  plane.  In  this 
case,  when  the  centre  of  the  shadow  is  at  c,  the  point  b  is  on  the 
meridian,  and  when  the  centre  is  at  c?,  the  point  a  is  also  in  this  case 
toward  the  east,  from  the  meridian.  The  node,  it  is  seen,  is  always 
distant  from  the  meridian,  in  the  direction  of  the  end  of  the  path 
which  is  farthest  from  the  pole. 

But  the  degrees  of  longitude  are  so  small  where  this  point  is 
formed  that  the  longitude  of  it  may,  without  error,  be  taken  as  that 
of  the  meridian  of  the  sun  at  noon. 

//!        or    ^  ±  180°. 

The  maximum  curve  always  lies  between  the  rising  and  setting 
curve  and  the  pole.  And  when  the  pole  is  elevated,  as  in  CHATJ- 
VENET'S  example,  and  the  shadow  passes  over  and  includes  the  pole, 
the  maximum  lies  near  the  node.  But  when  the  pole  is  depressed 
and  the  shadow  does  not  reach  the  pole,  the  maximum  lies  much 
farther  from  the  node  than  in  the  preceding  case — sometimes  as  much 
as  one  or  two  degrees.  This  is  quite  noticeable  in  the  eclipses  of 
1898,  July  18 ;  and  1899,  January  11. 

78.  Greatest  Eclipse. — CHAUVENET'S  chapter  has  no  mention  of 
this ;  but  under  these  words,  the  data  for  this  phase,  for  every  par- 


90 


THEORY  OF   ECLIPSES. 


78 


tial  eclipse,  is  given  in  the  Nautical  Almanac.     In  Art.  53  we  gave 
the  formulae  for  the  criterion,  and  in  Art.  52  for  the  time,  as  follows  : 


T  = COS  (M0  —  N) 


(110) 


FIG.  10. 


T=    T0  +   T  (111) 

In  a  partial  eclipse  the  extreme  times  will  lie  somewhat  closer 
together  than  in  a  larger  eclipse.     The  mean  of  the  two  terms  of  the 

beginning  and  ending  should  be  gotten 
closely  to  at  least  four  decimals  of  a  min- 
ute, on  account  of  its  effect  on  the  longi- 
tude. The  point  on  the  earth's  surface 
from  which  the  greatest  eclipse  is  visible 
must  lie  in  the  fundamental  plane,  since 
that  part  of  the  earth  is  nearest  to  the 
centre  of  the  shadow  ;  and,  as  its  name 
implies,  must  be  a  point  of  the  maxi- 
mum curve.  Hence,  the  formulae  in 
this  section  are  applicable  to  this  spe- 
cial case. 

In  determining  the  time  as  above  by 
the  condition  that  m  =  p  +  I  is  a  mini- 
mum, this  distance  is  CZ(~Fig.  10),  which 

line  is  consequently  perpendicular  to  the  path,  and  the  following 
changes  result : 


In  57 


cos     = 


or 


n 


(112) 


This  equation,  as  we  saw  in  Art.  58,  is  independent  of  the  extreme 
times,  and  is  general  for  the  whole  eclipse. 


In  59     r=N+<p=N±W 


(113) 


If  x  and  y  are  taken  at  the  time  of  the  middle  of  the  eclipse,  we 
must  have 


^±90° 


(115) 


The  general  condition  for  the  maximum  curve  (CHAUVENET,  i.,  p. 
476,  Art.  309)  is 


that  is, 


or     §=£+180° 


(116) 


78  MAXIMUM   CURVE.  91 

At  the  point  c?,  whence  the  shadow  crosses  the  line  CZ,  we  have 
the  angle  of  position,  §,  of  the  centre  of  shadow, 


Whence  M  =  E,  in  northern  hemisphere. 

M  =  180°  -f  E,  in  southern  hemisphere. 

Equation  110  may  be  put  under  another  form,  giving  rt  from  the 
time  of  conjunction.  At  this  time  M0  —  0,  m0  =  y^  and  we  also  have 
generally  n  cos  N=yf;  substituting  these  values,  we  obtain  the  time 
of  greatest  eclipse  from  the  time  of  conjunction 

*--*•  (117) 


which  can  be  used  as  a  check,  n  may  be  taken  from  the  computa- 
tion for  the  extreme  times,  and  the  other  quantities  are  in  the  eclipse 
tables. 

The  above  transformations  are  given  for  the  reader's  information, 
that  he  may  see  the  changes  in  the  several  quantities,  rather  than  for 
practical  use.  E,  however,  is  theoretically  correct  for  giving  the  lati- 
tudes and  longitudes,  but  as  it  is  given  only  to  the  tenth  of  a  minute 
in  the  table,  M  should  be  preferably  gotten  from  x  and  y,  especially 
as  ra  is  required  for  the  magnitude  ;  but  if  M  and  m  are  not  com- 
puted, m  may  be  interpolated  from  the  rising  and  setting  curve. 

The  formulae  for  greatest  eclipse  will  then  be,  approximately, 

M=E,        or    M  =  E  +  180°,  interpolated  from  the  tables.  (118) 
m  interpolated  from  R.  and  S.  curve.    (119) 

Or,  more  accurately,  for  the  time  T  equations  (110)  and  (111). 


rinJf-*,!  (120) 

cos  M  =  y.  ) 

tan  Yf  =  Pi  tan  r=  Pi  tan  M.  (121) 

?>!  sin  &  =  sin  Y'>  ^ 

?>!  cos  #  =  —  cos  Y'  sin  dlt  >  (122) 

sin  <PI  =  cos  Y'  cos  di.      ) 


tan?=--.  (123) 

«  =  ^  -  *  (124) 

Compute  with  five-place  logarithms  to  seconds  of  arc. 
The  reader  will  notice  that  M  =E  is  geometrically  shown  in  Fig. 
3,  the  quantities  forming  the  miniature  path  being  such  that  this  is 


92  THEORY   OF   ECLIPSES.  78 

parallel  to  the  real  path ;  and  when  this  figure  is  transferred  from  7* 
to  F,  the  two  perpendiculars  to  these  lines  form  one  straight  line. 

79.  Magnitude. — The  following  transformations  take  place  under 
the  conditions  of  the  previous  article  : 

In  (95)  since  M=E,      sin  <p  =  0. 

And  (97)  becomes  (Fig.  10)  A  =  m  —  p  =  ZC  —  Ze  =  Ce.     (125) 

The  distance  the  point  e  is  immersed  in  the  shadow  is 

de  =  I  —  A.  (126) 

The  point  /  of  the  umbral  cone  is  on  the  dividing  line  where  the 
sun  is  wholly  obscured,  so  that  the  magnitude  of  the  eclipse  is  the 
ratio  de  :  df;  and  as  Of,  the  radius  of  the  umbral  shadow,  is  negative 
for  a  total  eclipse,  df  will  be  the  algebraic  sum  I  -f-  llm  Hence,  we 
have  the  formula  for  the  degree  of  obscuration,  D,  of  CHAUVENET, 
generally  called  the  magnitude  in  the  Almanac. 

A  =  m-p.  (127) 

M-T&  (128) 

The  above  computation  gives  m;  and  py  being  the  radius  of  the 
earth,  can  be  taken  from  Table  IV.  Reduce  all  the  quantities  to 
natural  numbers  for  these  equations.  These  formulae  give  a  decimal 
of  the  sun's  diameter  taken  as  unity.  Because  when  M  becomes 
equal  to  unity,  the  sun  is  wholly  obscured  on  the  earth  at  e  (Fig. 
10).  The  magnitude  is  given  to  three  decimals  in  the  Almanac. 


SECTION  VIII. 
OUTLINE  OF  THE  SHADOW. 

80.  THIS  curve  shows  the  outline  of  the  penumbral  shadow  upon 
the  earth  at  any  moment.  The  times  are  here  assumed  at  pleasure 
for  the  centre  of  the  circle,  and  are  generally  taken  for  each  integral 
hour  successively,  giving  from  three  to  five  curves.  They  are  some- 
times called  Hour  Circles  (Figs.  2,  3,  and  8). 

The  Angle  Q. — There  is  a  mistake  in  CHAUVENET'S  text,  page 


,e 


80  OUTLINE   OF   THE  SHADOW.  93 

447,  Art.  292,  and  also  in  Fig.  42,  that  greatly  obscures  his  explana- 
tion of  the  angle  Q.  In  the  figure  the  line  -MiQ  should  not  pass 
through  the  centre  0.  M  is  the  centre  of  the  circle  of  shadow,  and 
Ol  is  any  point  on  this  circle.  The  whole  circle,  or  at  least  an  arc, 
might  well  have  been  drawn  in  perspective.  The  triangle  M^QN 
should  be  moved  a  little  to  the  right,  and  1/iQ  should  then  meet 
the  axis  in  some  point  between  O  and  Y.  The  figure  as  drawn  is 
correct  for  one  position  of  C19  but  is  not  general.  When  changed  as 
here  suggested,  the  text  will  not  apply  to  the  figure. 

A  modification  of  CHAUVENET'S  figure  39,  page  440,  will  explain 
this  angle.     In  Fig.  11,  let  we  be  the  fundamental  plane  and  the 
circle  below,  the  outline  of  the  shadow  on  the  plane. 
The  axis  of  the  cone  and  shadow,  the  line  joining  the         FlG- 11- 
centres  of  the  sun  and  moon,  is  perpendicular  to  the          S  Q 
fundamental  plane  at  all  times.     Suppose  a  plane  to  \/ 

be  passed  through  the  axis  of  the  cone,  it  will  pass  /A 

through  the  centres  of  the  sun  and  moon,  and  also  /    \ 

through  their  point  of  visual  contact.    If  it  also  passes  /       \ 

through  the  point  w,  an  observer  at  this  point  will          / 
see   the   western   limb    of  the   moon   tangent  to  the 
eastern  limb  of  the  sun,  the  moon  being  due  east  of 
the  sun.     An  observer  at  8  will  see  the  moon  like-         \i  \, 

wise  due  north  of  the  sun.      The  observer  having         I  J 

moved  90°,  the  moon  has  apparently  done  the  same ;          ^-jr^ 
and  generally,  from  whatever  position  the  observer 
is  at  on  this  circle  of  shadow,  the  moon  will  be  seen  on  the  opposite 
side  of  the  sun's  disk. 

Q  is  the  position  angle  of  the  centre  of  the  shadow  from  any  point 
on  the  circle.  At  8  this  angle  is  zero,  and  the  moon  being  seen  due 
north  of  the  sun,  the  position  angle  of  the  moon  on  the  sun's  disk  is 
also  0.  At  w  the  angle  is  90°  ;  the  moon  being  seen  due  east  of  the 
sun,  its  position  angle  on  the  sun's  disk  is  also  90°.  And  generally, 
Q,  the  position  of  the  centre  of  the  shadow,  is  at  all  times  equal  to 
the  position  angle  of  the  moon  on  the  sun's  disk.  The  angle  is  meas- 
ured from  the  north  point  of  the  sun's  disk  and  from  the  axis  of  Y 
on  the  earth  toward  the  east  as  positive. 

In  Art.  298  of  CHAUVENET'S  Astronomy,  page  459,  occurs  the 
following  equation,  not  numbered,  but  which  we  will  refer  to  as 
equation  (a)  : 

^2  =  cos2  /?  —  2  t'd  sin  p  cos  (  Q  —  Y)  —  O'd)2  (a) 

The  last  term,  being  the  square  of  a  very  small  quantity,  is  omitted. 


94 


THEORY  OF  ECLIPSES. 


80 


By  transposing  the  term  containing  ft  to  the  left  member,  com- 
pleting the  square,  and  then  extracting  the  root  of  both  members, 
wre  obtain — 

d  =  ±  cos  ft  —  i  sin  /3  cos  (  Q  —  Y)  (b) 

But  we  may  obtain  this  under  the  form  that  CHAUVENET  has  given 
by  replacing  ^  in  the  second  number  of  equation  (a)  by  cos  /5,  since 
cos  ft  =  £  exactly,  and  =  ^  nearly  ;  then  completing  the  square  as 
above  and  taking  the  root,  we  obtain 

Cj  =  ±  [cos  /3  —  i  sin  /5  cos  (  Q  —  y)]  (c) 

As  (Q  —  f)  gives  any  point  of  the  outline  of  shadow,  if  we  let  it 
represent  two  points  diametrically  opposite  one  another,  it  will  take 
the  double  sign  ±,  which  will  then  reduce  equation  (b)  to  the  form 
(c).  CHAUVENET  makes  use  of  the  latter  equation,  but  it  is  unim- 
portant, since  it  is  used  only  for  e  and  to  determine  the  sign  of  £x ; 
and  this  he  ascertains,  in  the  next  following  paragraph,  simply  by  a 
process  of  reasoning. 


81.  General  Formula. — 


sin  /?  sin  f  =  x  —  I  sin  Q  =•  a     ' 

v        I  cos  Q 
sm  /?  cos  r  =  — =  & 


P\ 


ft 


I.  4 


sin  V1 
cos  09  +  e) 
a  +  iCi  sin  Q  j 
b  +  %  cos  Q  ) 


(129) 

(130) 
(131) 
(132) 


For  greater  accuracy,   instead   of  the    previous  group,   take   the 
following : 


II. 


sin  Q  =  ? 


(133) 
(134) 

(135) 

C,=  cos/9'  (136) 

*  This  quantity  may  be  approximately  taken  from  Table  X.    An  example  of  its 
use  is  given  in  Section  X. 


sin  1" 

e'  =  (c?t  —  d.2}  COS  Y 
sin  /5'  sin  Y'  =  a  +  *j°a  cos  (?  + 
sin  /9'  cos  Y'  =  b  ~ 


Pi 


81  OUTLINE   OF   THE   SHADOW.  95 

Whichever  one  of  the  two  above  groups  has  been  used,  then 
proceed  as  follows : 

c8inC=7!)  m7N 

eeosC.-Cj 

cos  <?i  sin  $  =  £  ~\ 

cos  ^  cos  '9  =  c  cos  (  C  -f  dj  I  (138) 

sin  <pv  =  c  sin  (  C  +  dj)  J 

tan^=     taii^  (139) 

a*  =  /*!  -  *  (140) 

These  formulae  are  rigorous  and  will  bear  five-place  logarithms 
and  close  work ;  the  computer  may  take  his  choice  of  the  two 
groups,  which  one  to  use,  according  to  the  accuracy  required. 
Formulae  (134)  and  (135),  it  will  be  noticed,  require  the  quantities 
d2  and  p2,  which  are  not  usually  computed  for  an  eclipse.  They  are 
gotten  from  formula  (51),  Art.  34.  The  equations  are  to  be  com- 
puted for  a  series  of  assumed  values  of  §,  so  that  the  first  thing  to 
be  done  is  to  find  the  limits  of  Q.  The  time  for  which  the  outline 
is  to  be  computed  must,  of  course,  be  previously  assumed.  The  sim- 
plest way  to  find  Q  is  from  a  plot  of  the  eclipse  similar  to  these  here 
given.  Thus,  in  Fig.  8,  Plate  V.,  the  centre  of  the  10A  curve  bears 
from  the  southerly  end  about  -f-  20°,  and  the  points  extend  round 
the  circle  to  -f-  220°,  in  which  direction  the  centre  bears  from  the 
northerly  end.  Then  the  series  of  values  is  to  be  assumed  every  5° 
or  10C,  as  circumstances  may  require.  For  the  convenience  of  the 
computer  Table  IX.  is  appended,  giving  the  sines  and  cosines  for 
every  5°  of  the  circle.  The  outlines  for  8A  and  9A  (Fig.  8)  are  entire, 
and  Q  will  give  values  throughout  the  whole  circle.  The  small  quan- 
tity e,  formula  (130),  can  be  taken  approximately  from  Table  X.  If 
a  value  should  be  assumed  for  Q,  for  which  there  is  no  point  of  the 
curve,  it  will  show  itself  by  giving  a  value  of  sin  J3  >  unity.  Sin  /9 
is  always  positive,  also  cos  /9,  as  the  latter  is  the  height  of  the  point 
on  the  earth's  surface  above  the  fundamental  plane ;  and  as  there  can 
be  no  eclipse  below  the  plane,  ft  will  be  between  0°  and  -f-  90°.  The 
angle  C  is  always  less  than  90°,  but  may  be  either  +  or  — . 

82.  Formulae  of  such  rigor  as  these  above  given  will  seldom  be 
required  for  this  curve,  though  it  is  the  most  important  of  all  the 
penumbra!  curves.  The  only  case  in  which  these  accurate  formulae 
would  be  required  is  probably  a  drawing  on  an  enlarged  scale  show- 


96  THEORY   OF   ECLIPSES.  82 

ing  the  path  of  the  shadow  over  a  tract  of  country.  The  outline 
curves  in  this  case,  computed  for  short  intervals  of  time,  say  5  or  1 0 
minutes,  or  closer  perhaps,  will  serve  to  warn  observers  of  the 
approximate  times  of  beginning  and  ending,  and  that  is  all  he 
requires. 

83.  The  following  criterion  is  given  by  CHAUVENET,  though  it 
will  seldom  or  never  be  required  by  the  practical  computer : 

e  sin  ( Q  —  E}  <  d / sin  Q,  Eclipse  is  beginning, ) 
e  sin  (  Q  — -  E)  >  Ci  /  sin  §,  Eclipse  is  ending.       J 

No  example  of  the  use  of  these  formulse  for  outline  is  appended, 
for  the  reasons  given  in  the  next  paragraph. 

84.  Approximate  Formulae. — The  formulse  above  given  by  CHAUVE- 
NET,  being  carried  to  their  utmost  exactness,  the  reader  may  perhaps 
expect  some  suggestion  as  to  approximate  formulse ;  but  this  is  de- 
ferred to  Section  X.,  on  the  Northern  and  Southern  Limiting  Curves, 
in  which,  when  the  angle  Q  is  determined  (which  is  here  assumed), 
the  formulse  are  the  same  precisely.     And  as  that  curve  is  of  much 
less  importance,  it  is  a  suitable  place  for  repeating  these  formulse 
with  sufficient  exactness  for  ordinary  use. 

85.  Geometrical  Illustration. — In  Fig.  8,  Plate  V.,  the  coordinates 
of  the  centre,  TQ,  of  the  9A  curve  are  x  and  y.     The  coordinates  of 
the  point  m,  referred  to  the  centre  of  the  shadow,  are  I  sin  Q  and 
I  cos  Q.    Hence,  the  coordinates  of  this  point  referred  to  the  axes  are 
x  —  I  sin  Q  and  y  —  I  cos  Q. 

Also  the  same  point  referred  to  the  axes  by  its  bearing  7-  from  CZ, 
and  distance  Zm,  which  is  here  called  sin  /9,  are  sin  /9,  sin  y,  and  sin  ft 
cos  ft.  Hence,  we  have 

sin  /?  sin  y  =  x  —  £  sin  §  =  a  =  £ 

sin  /?  cos  Y  =  y  —  I  cos  Q  =  b  =  rji 

It  follows  from  this  that  the  third  coordinate  is  £1  =  cos  ft ;  of 
which  the  reader  may  not  need  an  explanation ;  however,  it  is  as  fol- 
lows :  It  results  upon  the  general  equation  of  the  sphere  (CHATTVE- 
NET,  Art.  298,  p.  458,  equation  (499)). 

£'  +  ^  +  d1  =  1  (142) 


85  OUTLINE   OF   THE  SHADOW.  97 

substituting  in  this  the  values  of  £  and  %  from  the  previous  equa- 
tion, sin2  Y  +  cos2  f  cancels  out,  and  we  have  left 

sin2  /?  +  d2  ==  1 
whence  Ci2  =  1  —  sin2  /?         or         d  =  cos  /?. 

The  small  term  e  is  added  to  ft,  to  take  account  of  the  diminished 
radius  of  the  penumbral  shadow  as  we  approach  the  fundamental 
plane  in  a  total  eclipse.  In  the  equation 


sinl" 

i  —  tan  /,  the  angle  of  convergence  of  the  cone  ;  tan/  is  the  amount 
of  this  at  the  height  ^  above  the  fundamental  plane.  And  being 
added  to  the  angle  ft,  when  ft  =  0°,  f  is  at  the  point  Z  (Fig.  8),  and 
its  value  not  changed,  because  the  surface  of  the  sphere  is  parallel 
to  the  fundamental  plane.  When  ft  =  90,  it  affects  the  angle,  but 
not  the  cosine  ;  and  £  is  not  affected  at  this  point,  because  the  diver- 
gence of  the  cone  is  0.  The  correction,  it  is  seen,  very  well  answers 
its  purpose  when  applied  to  the  angle  ft,  rather  than  in  any  other 
way.  Moreover,  at  the  point  c  (Fig.  13),  where  Q  =  f,  e  will  be 
positive,  which  increases  the  angle,  but  diminishes  £  ;  which  is  cor- 
rect, as  the  convergence  of  the  cone  diminishes  the  radius  toward 
the  centre  of  the  cone.  At  the  point  d,  where  Q  =  180  +  f,  the 
reverse  is  the  case.  The  correction  is  not  quite  exact  at  inter- 
mediate points,  so  that  a  second  quantity  is  added  in  the  rigorous 
equations. 

86.  The  geographical  position  of  the  point  is  completely  deter- 
mined by  these  three  coordinates.  In  Fig.  8  let  ra  be  the  point  on 
the  9A  curve.  Pass  a  vertical  plane  through  this  point  perpendicu- 
lar to  the  axis  of  X  at  n,  and  revolve  it  down  in  the  principal  plane  ; 
ra  and  n  being  in  the  axis  of  revolution,  will  remain  unchanged  ;  the 
points  of  the  sphere  vertically  above  them  will  fall  at  p  and  q  respec- 
tively on  the  semicircle  of  which  mn  produced  is  the  diameter  ;  the 
point  r  of  the  equator  will  fall  at  s. 

Then  pnq  =  C 

pn  —  c 

qns  =  d     or     di 
pns  =  C  4-  dl 


98 


THEOKY  OF   ECLIPSES. 


86 


And  the  point  m  is  now  referred  to  the  equator  instead  of  the 

plane  XZ,  so  that  the  lati- 
tude now  soon  follows. 

The  relations  of  the  quan- 
tities following  being  in 
space,  are  more  clearly  seen 
from  Fig.  1  2,  annexed,  which 
is  a  perspective  view  of  the 
eighth  part  of  the  sphere. 
The  reference  letters  being 
the  same  as  in  Fig.  8, 


Plate   V.,    the    points 
readily  be  identified. 


EB  is  the  equator.     P  is  the  pole. 


can 


pn  =  c. 

pmr  —  c  cos  (  C  4-  di). 
m'n  =  c  sin  (  C  +  d^). 
Also  Ppt  is  the  meridian  of  the  point  p. 

pt  =  q>9  the  arc  of  latitude. 
Ept  =  #,  the  hour  angle. 
Then  Ov  =  cos  <p. 

On  =  cos  <p  sin  $  =  £. 
vn  =  pmf  =  cos  <p  cos  #  =  c  cos  (  (7  +  d  i). 
pv  =  771/n  =  sin  <p  =  c  sin  (0  -f  d^). 

Small  quantities  must  necessarily  be  omitted  in  the  above  illustra- 
tions, and  d  =  dl9  so  far  as  can  be  seen  on  a  drawing  of  the  size  of 
Fig.  8. 

87.  Dr.  Hill's  Formulae  for  Outline  Curves.  —  The  formula  are  simi- 
lar to  those  prepared  by  him  for  projecting  the  Transit  of  Venus  in 
1874.* 

NOTATION  FOB  THIS  ARTICLE. 
h  =  altitude  of  the  sun  above  the  horizon,  nearly. 
=  altitude  of  the  point  of  contact  above  the  fundamental 

plane,     h  is  always  less  than  90°. 
0  =  parallactic  angle  of  point  of  contact. 
=  where  projected  on  the  fundamental  plane,  the  angle 

of  point  of  contact  from  the  axis  of  Y. 
P  =  radius  of  the  terrestrial  spheroid. 

*  Papers  on  the  Transit  of  Venus  in  1874.  Keport  of,  by  Mr.  George  W.  Hill, 
Washington,  1872. 


87  OUTLINE   OF   THE  SHADOW.  99 

The  fundamental  equation  in  the  theory  of  eclipses  (CnAUVENET, 
i.,  p.  449,  equation  (490))  is, 

(/  -  iC)2  =  (x  -  £)'  +  (y  -  T?)'  (143) 

In  the  small  term  place  £  a  mean  value  =  J. 


Place  x  =  m  sin  M        y  =  m  cos  M  C144\ 

(f-  £i)'  =  ™  '        ' 


Now  A  being  the  altitude  of  the  place  above  the  fundamental 
plane,  we  have 

C  =  p  sin  h 
p  -f  -f  =  f  cos2  h 
?  +  V  +  C3  =  />2  =  1  nearly 

(7  -  i  i)2==  wi2  -  2(^  +  yri  +  /o1  cos3  A 


But  £  =  /o  cos  A  sin  0 

j]  =  p  cosh  cos  0 
And  place          />  =  1 

and  x£  -\-yy  =  m  sin  M  '  />  cos  h  sin  0  +  m  cos  M'  p  cos  A.  cos  0 

=  m  cos  h  cos    0  — 


fa       arx 
cos  (<9  —  M  ) 


2m  cos  A 


the  small  term  ^i  in  the  above  equations  being  omitted.  This 
equation  completely  determines  the  point.  The  sign  of  cos  (8  —  M) 
is  determined  ;  it  is  usually  positive  ;  but  when  the  curve  goes  over 
the  zenith  for  some  few  points  the  angle  is  obtuse.  (0  —  M)  has 
two  values,  positive  and  negative.  It  is  analogous  to  ^  in  CHAUVE- 
NET'S  formulae  here  given,  No.  78  of  the  Rising  and  Setting  formulae. 
It  is  the  angle  at  the  centre  of  the  sphere  to  any  point  of  the  curve. 
In  Fig.  13,  A,  if  (6  —  M  )  =  0,  cos  (6  —  M  )  =  ±  1,  then  equation 
(147)  gives 

±  (m  rb  cos  A)  =  ±  I. 

When  the  outline  touches  the  horizon  (Fig.  13,  A),  which  is  the  most 
usual  case, 

cos  h  —  m  —  I 

the  superior  limit  a,  and  there  is  no  inferior  limit,  since  it  falls  off 
the  sphere. 


100  THEORY  OF   ECLIPSES. 

% 

When  the  circle  does  not  touch  the  horizon  (Fig.  13,  B\ 

cos  h  =  m  —  I,  the  superior  limit  c 
cos  h  =  m  +  I,  the  inferior  limit  d 

When  the  curve  passes  over  the  zenith  (Fig.  13,  C), 

cos  h  =  m  —  /,  still  the  superior  limit  e 
cos  h  =  m  +  I,  the  inferior  limit  / 

FIG.  13. 


87 


In  the  latter  case,  however,  m  —  /is  negative,  and  for  a  portion  of 
the  curve  adjacent,  6  —  M>  90°,  and  becomes  — 180°  at  the  point  e. 


88.  Hence  we  have  for  practical  use, 
Limits  of  h  when  the  curve  touches  the  horizon, 

h  =  0,         or     cos  h  =  1 

cos  h  =  m  ~» 

When  the  curve  does  not  touch  the  horizon, 

cos  h  =  m  H 
cos  h  =  m  •* 

m  sin  M  =  x 

m  cos M=y 

mz  —  P  -f  cos2  ^         , 
cos  (0  —  Jf )  = ~ sec  h. 


(145) 

(146) 
(147) 


88  OUTLINE   OF   THE  SHADOW.  101 

And  for  the  geographical  positions, 

cos  <p  sin  &  =  cos  h  sin  6  ^ 

cos  <p  cos  #  =  cos  d  sin  h  —  sin  d  cos  h  cos  6  V  (148) 

sin  <p  =  sin  d  sin  A  -j-  cos  d  cos  ft  cos  0  J 

As  d  varies  but  a  few  minutes  during  an  eclipse,  it  may  be  taken 
as  constant,  as  also  1.  M  and  m  may  be  taken  from  the  rising  and 
setting  curve  for  the  eclipse  hours. 

It  will  be  noticed  that  the  following  are  constant  for  all  eclipses : 
log  sin  A,  log  cos  A,  log  sec  A,  and  cos2  h  natural  numbers.  They  may 
be  prepared  beforehand,  and  are  given  in  Table  XI.  for  every  five 
degrees. 

The  following  are  constant  for  one  eclipse  : 

log  sin  d,  log  cos  d,  I*  in  numbers ; 

also  the  factors, 

(1)  =  log  (cos  d  sin  A),  (2)  =  log  (sin  d  cos  A), 

(3)  =  log  (sin  d  sin  h\  (4)  =  log  (cos  d  cos  k). 

And  the  following  are  constants  for  each  curve  : 

M,      //u      m2  in  numbers    or  (m2  —  Z2),       (w  —  Z)       log  m. 

These  can  be  prepared  beforehand  on  the  lower  edge  of  a  slip  of 
paper — the  general  constants  for  every  5°  or  10°  of  h. 

Compute  with  four-place  logarithms  to  minutes  for  the  eclipse 
hours.  (6  —  M\  having  the  double  sign,  gives  two  points  to  the 
curve.  The  inferior  limit  gives  but  one  point  for  which  6  =  M. 
The  superior  limit  when  it  exists  also  gives  but  one  point,  for  which 
6  =  M-{- 180°.  Instead  of  finding  tangent  <p,  it  is  better  to  find  the 
angle  from  its  sine  and  cosine,  and  their  agreement  with  the  same 
angle  is  a  considerable  check  upon  the  accuracy  of  the  work.  If 
the  computer  has  confidence  in  his  work,  he  may  find  <p  from  its 
cosine  alone  when  the  point  is  not  too  near  the  equator.  If  the  sine 
and  cosine  agree  within  10  minutes,  the  point  may  be  passed  as  cor- 
rect, or  at  least  until  plotted  on  the  chart,  for  the  ink  line  to  be 
drawn  through  this  point  will  be  about  10  or  15  minutes  of  arc  in 
width.  A  plot  of  the  eclipse  similar  to  those  here  given  will  be 
found  very  useful  in  roughly  checking  the  work,  or  deciding  the 
quadrant  to  be  taken.  If  a  scale  be  made  of  cosines  to  the  radius 
of  the  sphere,  any  point  can  at  once  be  located  on  the  hour  circles. 


102  THEORY   OF   ECLIPSES.  89 

89.  Geometrical  Illustration. — Not  much  is  required  under  this 
heading.  Equation  (147)  is,  after  all  the  transformations,  very 
simple.  In  the  common  formula  for  a  plain  oblique  triangle, 

a2  =  62  +  c2  —  26c  cos  A 
or  cos  A  — — 


If  we  apply  this  to  the  triangle,  Zrq  (Fig.  13),  which  lies  in  the 
fundamental  plaue,  calling  A  the  angle  at  Z,  equation  (147)  results 
at  once,  for  Zq  =  cos  A,  rq  =  I,  Zr  =  m.  The  angle  PZr  =  M  is 
known ;  PZq  =  6  is  required,  which  is  given  by  d  —  M  =  qZr  =  A 
of  the  above  general  formula. 

To  proceed  further :  if  in  the  general  equation  sin  \a  =  ^ — > 

2 

we  substitute  the  value  of  cos  (6  —  M)  from  Dr.  HilFs  formula  (147), 
we  get,  after  reducing, 


8n 


4pm  *  4pm 

which  is  CHAUVENET'S  equation  (78)  deduced  from  Dr.  HilPs. 
Equations  (148)  are  of  the  usual  forms,  the  point  p  being  referred 
first  to  the  fundamental  plane  and  axes,  then  the  axes  revolved 
through  the  angle  d,  which  is  the  inclination  of  the  equator  to  the 
plane  XZ. 

In  the  example  following  are  a  few  references  to  Fig.  8  in 
explanation.  The  reader  will  readily  see  that  all  these  quanti- 
ties, lines,  and  angles  can  be  measured  on  the  drawings  given  to 
scale  in  the  plates. 

90.  Example.  —  Outline  by  Dr.  HilPs  Formulae.  The  several  con- 
stants for  this  eclipse  and  for  the  9A  curve  are  given  with  the  exam- 
ple. It  will  be  necessary  almost  to  write  these  off  on  a  slip  of  paper, 
and  to  follow  the  formulae,  to  understand  the  example,  for  they  are 
not  given  there.  The  example  stands  just  as  it  does  on  my  com- 
puting sheet,  except  that  it  should  form  one  column.  Constants  are 
written  once,  and  I  avoid  as  much  as  possible  repeating  them.  In 
Fig.  8,  Plate  5,  if  a  circle  of  altitude  be  drawn  whose  radius  is  cos  h, 
it  will  strike  the  9A  outline  in  two  points,  i  and  j9  which  are  those  of 
the  example.  The  references  (2)  and  (4)  in  the  precept  to  the  exam- 
ple refer  to  those  constants  given  in  the  margin.  (1)  and  (3)  are  to  be 
used  from  a  slip  of  paper.  The  natural  numbers  are  used  for  m2  —  I2, 


90 


OUTLINE   OF   THE  SHADOW. 


103 


and  this,  added  to  cos  A,  natural  numbers,  gives  the  numerator  ;  then 
pass  to  logarithms  (0  —  M).  6  and  M  can  be  measured  on  Fig.  8, 
0  =  CZi  and  CZj,  and  corresponds  to  7-  of  CHAUVENET'S  formulas. 
The  quantities  in  equation  M  and  log  m  are  taken  from  the  rising 
and  setting  curve  generally,  but  the  example  given  was  not  the  9A 
curve.  However,  as  this  is  the  epoch  hour,  these  quantities  may  be 
found  in  Article  54,  Extreme  Times  Generally. 

OUTLINE  CURVES,  HILL'S  FORMULA. 
EXAMPLE,  TOTAL  ECLIPSE,  1904,  SEPT.  9. 


Constants  for  all  curves  for  this  eclipse,  inter- 
mediate values  being  omitted. 


h. 

0) 
(2) 
(3) 
(4) 


Pi 
M 


h. 

60 

60 

(136)  0 

+  68    15 

-f  240  7 

inter- 

sin 

+  9.9679 

—  9.9380 

cos 

+  9.5689 

—  9.6974 

80 

cos  $  sin  # 

9.6669 

—  9.6370 

9.9916 

(2)cos0 

-f  8.2291 

—  8.3576 

8.2009 

I  —  I 

B  1.7066 

A  1.5781 

8.9546 

A  1.6980 

B  1.5894 

9.2379 

COS  0  COS  # 

+  9.9271 

+  9.9470 

tantf 

9.7398 

9.6900 

•& 

+  28   47 

—  26     6 

cos 

9.9427 

9.9533 

cos  <j> 

-f  9.9844 

+  9.9937 

(4)cos0 

-f  9.2661 

—  9.3946 

I  —  I 

A  0.3674 

B  0.4959 

B  0.5225 

A  0.3289 

60 

sin  0 

+  9.4212 

—  9.2276 

-f  0.015! 

*            c» 

+  15   17 

9     43 

4-8.198, 

'  (137)  w 

f  106   55 

-f  161    48 

20  40  60 

+  9.5323  9.8063  9.9357 

+  8.9342  8.8455  8.6602 

-f  8.4953  8.7693  8.8987 

+  9.9712  9.8825  9.6972 

Constants  for  this 
curve. 

135°  42'        9*  curve 
154    11       for  h  =  60° 

log—  0.3515 

h. 

(mj_p)_  0.2342  (135)  numerator  +0.0158 
log 

cos  (0— Jf)  +  8.8512 
(0— M)      =F85   56 

It  may  be  added  that  the  omission  of  t'f  or  J  i  in  equation  (147) 
has  exactly  the  same  effect  as  omitting  the  quantity  e  (equation 
(130))  from  the  approximate  formulae  referred  to  in  Article  84,  for 
they  both  take  account  of  the  decreased  radius  of  the  hour  circles  as 
the  points  rise  above  the  fundamental  plane. 

The  Rising  and  Setting  Curve. — Article  64  is  derived  from  the 
above  formulae  simply  by  the  condition  that  h  =  0.  All  the  formula 
are  simplified  by  the  point  being  in  the  fundamental  plane,  and 
(1  —  I2)  is  the  only  constant. 


104  THEORY   OF   ECLIPSES.  91 


SECTION   IX. 

EXTKEME  TIMES,  NOETHEEN  AND  SOUTHEEN  LIMITS  OF 
PENUMBEA. 

91.  IN  this  section  CHAUVENET  remarks  that  he  has  not  followed 
BESSEL,  but  has  given  formulae  better  suited  to  practical  use.  The 
formulae  in  full  are  as  follows  : 

m  sin  M  =  XQ  qp  I  sin  EQ j  Cl 49") 

m  cos  M  =  2/0  =F  I  cos  E0  j 

nsmjy=ab'=F  *V'l 

j  (150) 

»ooBjNr«jfo'qF±eft// 

G        J 
sin  ^  =  m  sin  (Jlf  —  N)  (151) 

cos  ^        m  cos  (M  —  N)  f 

n  n 

T=T0  +  T  (153) 

r  =  N+<!>  |  (154) 

tan  Yf  =  Pi  tan  7-  j 
cos  ^  sin  $  =  sin  f  "| 

cos  ^i  cos  &  =  —  cos  p'  sin  d^  V  (155) 

sm  ?>i  ==  cos  T'/  cos  c?!      J 

tan  ^  =  — -^-  (156) 

vr^72 

o>  =  MI  -  *  (157) 

m  i  .       7-r  (  For  Northern  Limits,  take  the  lower  sign. 

TaklDg^aCUte'  I  For  Southern      "  "        upper" 

Compute  with  four-place  logarithms  to  minutes,  and  with  but  one 
approximation.  The  compression  may  be  neglected  in  this  curve ; 
then 


For  these  points  Q  =  E,  which  is  here  taken  as  acute,  which  gives 
rise  to  the  double  sign,  for  in  fact  E  has  two  values  differing  by 
180°. 

The  quantities  with  subscript  zero  are  to  be  taken  for  the  epoch 
hour ;  I  also  may  be  taken  as  constant,  and  if  the  above  approxima- 
tions are  made,  d  may  be  used  instead  of  d^  and  taken  as  constant  at 
the  epoch  hour.  ^,  as  in  similar  equations,  has  the  two  values,  acute 
and  obtuse,  giving  the  +  and  —  values  to  the  cosine. 


91  EXTREME   TIMES— LIMITS   OF   PENUMBRA.         105 

The  equations  for  N  are  the  hourly  motions  of  the  equations  for 
M.  If  we  substitute  in  the  first  equation  e  sin  E  =  bf,  we  have  in 

the  right  member  x0  q=  I  —  •     Now  the  hourly  motion  of  x  is  x',  and 

6 

of  bf  it  is  b",  which  gives  the  first  equation  for  N,  e  being  taken  as 
constant. 

CHAUVENET  states  in  Article  313,  at  the  bottom  of  page  485,  that 
e  may  be  taken  as  constant ;  it  is  because  it  has  but  little  eifect,  occur- 
ring only  in  a  very  small  term.  If  e  is  interpolated  for  the  two  times 
of  beginning  and  ending,  it  will  be  found  that  the  two  values  are 
quite  or  nearly  the  same,  since  e  has  a  maximum  or  minimum  at  the 
middle  of  the  eclipse. 

These  times  are  not  of  much  importance,  no  figures  are  given,  and 
they  are  used  merely  for  the  geographical  positions  to  check  the 
other  curves.  <p  should  be  looked  out  from  both  sine  and  cosine  as 
a  check. 

The  middle  of  the  eclipse  for  these  times  is  given  by  the  formula 

explained  in  Article  52. 

T==__msm(M-N)  (159) 

T^T0-r  (160) 

This  may  not  agree  exactly  with  that  found  for  beginning  and 
ending,  but  should  not  vary  much  from  it.  They  agree  very 
closely  in  this  eclipse,  but  it  is  a  mere  chance  and  nothing  more. 
The  formulae  are  so  similar  to  several  of  those  already  given  and 
explained  that  they  need  no  further  comments,  especially  as  the 
extreme  points  have  already  been  explained  in  Article  72  of  the 
maximum  curve. 

92.  Example. — This  comprises  the  whole  work  for  the  times,  only 
one  approximation  being  made ;  but  the  latitudes  and  longitudes  are 
omitted,  since  the  formulae  are  the  same  as  those  used  in  several 
examples  on  former  pages,  especially  for  the  outline  curves.  Log 
(1  :  e)  in  this  example  is  0.2364.  The  formulae  numerically  are  the 
same  for  beginning  and  ending  except  when  the  double  sign  gives 
two  values.  It  is  well  to  place  the  sign  at  the  head  of  the  columns ; 
for  the  north  points  the  terms  of  the  first  two  formulae  are  added, 
and  for  the  south  points,  subtracted.  Addition  and  subtraction  loga- 
rithms are  very  convenient  here,  since  the  difference  of  the  logarithms 
here  noted  as  I  —  I  is  the  same  for  both,  being  called  A  if  the  terms 
are  numerically  added,  and  B  if  they  are  numerically  subtracted  ; 
cos  <p  is  minus  for  beginning  and  plus  for  ending. 


106 


THEORY   OF   ECLIPSES. 


92 


EXTREME  TIMES,  N.  AND  S.  LIMITS,  PENUMBRA. 
EXAMPLE,  TOTAL  ECLIPSE,  SEPTEMBER  9. 


N.  (+). 

s.  (-). 

N.  (+). 

s.  (-). 

Formula  |logZ        +9.7264 
j.'iy      ) 

150 
log  (1:  e)0.2364 

log(M 

+9.9628 
+8.1268 

sin  E0     +9.4803 

c" 

—7.6182 

cosjEo     +9.9792 

150 

V 

+9.7465 

149 

x            +8.9866 

(I  :  e)  b" 

+8.0896 

lsinE0  +9.2067 

I  —  I 

A  1.6569 

5 

I  —  I      ^10.2201 

5 

5        95 

A        97 

50.4249 

A  9.8195 

nsin  N 

+9.7560 

+9.7368 

m  sin  M  +9.4115 

—8.8061 

150 

yj 

—9.2380 

149 

2/0           —9.3018 

(1  :  e)  c" 

—7.5810 

lcoaE0  +9.7056 

I  —  I 

A  1.6570 

5 

I  —  I       50.4038 

A 

5        95 

A        97 

A  0.1858 

50.5483 

n  cosjY 

—9.2475 

—9.2283 

m  cos  .M  +9.4876 

—9.8501 

150 

tanJV 

0.5085 

0.5085 

149 

tan  M    +9.9239 

8.9560 

N 

+107    14 

+107    14 

M      +40   1 

—174   50 

sin 

9.9801 

9.9801 

cos             9.8842 

9.9982 

log(lsn) 

+0.2241 

+0.2433 

log  m      +9.6034 

+9.8519 

OnN 

Curve.                           On  S.  Curve. 

151 

M—  N        —67    14 

—28.2    4 

sin  (     )         —9.9648 

—9.9903 

cos  (    )         +9.5871 

+9.3202 

sin  V             —9.5682 

—9.8422 

V                 —158    17 

—21   43      —135   57 

—44   3 

152 

cos  ^             —9.9680 

+9.9680        —9.8566 

+9.8566 

log  (1)          —0.1921 

+0.1921        —0.0999 

+0.0999 

(2)          +9.4146 

+9.4154 

Nos.  (1)        —1.5563 

+1.5563        —1.2587 

+1.2587 

Middle. 

—(2)        —0.2598 

—0.2598        —0.2603 

—0.2603 

r  -0.260 

T                   —1.816 

+1.297          —1.519 

+0.998 

T 

f  8.740 

J. 

\  8h  44m.4 

153 

T  f                  7.184 

10.297             7.481 

9.998 

I                  7  11.04 

10  17.82          7  28.86 

9  59.88 

The  several  times  we  have  now  computed  for  the  eclipse  may  now 
be  compared  in  the  following  manner.  The  central  times  can  also 
be  included,  as  here  shown,  after  they  have  been  computed. 

The  internal  contacts,  not  being  computed,  are  not  sufficiently 
exact  to  be  included  with  the  other  quantities.  The  differences 
throughout  this  whole  eclipse  are  much  more  even  than  usually 
found  among  eclipses. 


92      N.   AND  S.   LIMITING  CURVES   OF   PENUMBRA.     107 


TOTAL  ECLIPSE,  1904,  SEPTEMBER  9. 

Art.  54,  First  Contact  (Beginning),  6A     7m.8     n  55  2 

"  107,  Central  Eclipse         "  7       3  .0     Q     8'Q 

"     92,  Northern  Limit        "  7     11.0     Q  17'g 

"     92,  Southern  Limit        "  7     28.9     -,-,5*5 

"  107,  Middle  of  the  Eclipse,  8     44  .4     1  lg'5 

"     92,  Southern  Limit  (Ending),  9     59  .9     Q  17'9 

"     92,  Northern  Limit         "  10     17  .8           ?'9 

"  107,  Central  Eclipse         "  10     25  .7     0  55'2 

"     54,  Last  Contact             "  11     20  .9 

"     55,  Second  Contact  (Internal),  7     58.  (by  scale) 

"     55.  Third  Contact           "  9     29.        " 


SECTION    X. 

NOETHEKN  AND  SOUTHEKN  LIMITING  CURVES  OF  PENUMBRA. 
93.  CHAUVENET'S  formulae  in  full  are  as  follows  : 

For  Limits  of  Q,  tan  v  =  $-  (161) 

tan  0  =  tan  (45°  +  v)  tan  J  E  (162) 

Limits    I  S*  curve'  ^  between  E  and  *  E  +  #  I      n  6T> 

'   (  N.  curve,  Q  between  180°  -f  E  and  180°  +  J  E  +  <p  ) 

For  §,  sin  /5  sin  ^  =  «  —  I  sin  §  =  a  j  (164) 

sin  /?  cos  y  =  y  —  I  cos  §  =  6  ) 

tan  *>*=-£-  cos  /?  (165) 

6 

tan  (  §  -  }  E)  =  tan  (45°  +  /)  tan  *  #  (166) 

For  the  geographical  positions,  equations  (129-140)  in  full,  Article 
81,  for  Outline  of  Shadow.  By  omitting  repetitions,  these  may  be 
approximated  as  follows : 

sin  ft  sin  7-  =  x  —  /  sin  Q  =  £  ) 
sin  ft  cos  Y  =  y  —  I  cos  Q  =  y  ) 

£=ico8«?-r)    orTableX.        (168) 

sin  I" 

C  =  cos  (f  +  e)  (169) 

csinC  =  >n  (170) 

c  cos  (7  =  C  j 


THEORY  OF   ECLIPSES. 


cos  <?i  sin  #  =  $ 
cos  y>i  cos  #  =  c  cos  (  (7  -f 
sin  <f>l  =  c  sin  (  C  -f  c?3 

tan  CP, 
tan  <p  = — 


108  THEORY  OF   ECLIPSES.  93 

(171) 

(172) 

•  -  A  -  «  (173) 

These  formulae  can  still  be  greatly  simplified,  especially  the  method 
of  finding  §,  which  comprises  nearly  half  the  work  of  the  formulae ; 
and  always  seemed  to  the  author  very  unsatisfactory ;  though,  per- 
haps, in  a  strictly  mathematical  sense  it  may  be  the  best  that  can  be 
devised. 

94.  The  Limits  for  Q.— This  is  illustrated  in  Fig.  14,  the  Total 
Eclipse  of  1878,  July  29,  which  is  the  next  successive  eclipse  in  the 
series  to  CHAUVENET'S  example,  and  thus  very  similar  to  the  figure 

which  his  example 
would  give.  The  eclipse 
hours  are  here  num- 
bered ;  and  above  the 
base  line,  by  the  scale 
of  degrees  in  the  mar- 
gin, the  angle  E  is 
shown ;  the  points  being 


FIG.  14. 


so' 


-40 


30' 


11 


la  connected  give  nearly  a 

right  line.   By  formulae 

(161)  and  (162),  the  superior  limit  for  §,  J  E  +  <p,  is  likewise 
shown.  Between  these  limits,  on  the  hour  lines,  we  are  to  assume 
values  for  Q  for  the  eclipse  hours — a  rather  uncertain  thing  to  do, 
when  it  is  considered  the  true  values  of  Q  will  lie  on  the  curved 
line  between  these  limits.  Unless  we  know  the  form  of  this  curve, 
two  approximations  maybe  necessary  with  formulae  (164-166)  to  find 
Q  with  sufficient  exactness.  It  is  seen  that,  as  CHATJVENET  states, 
Q  =  E,  the  lower  limit,  at  the  beginning  and  end  of  the  curve ; 
and  at  the  middle  approaches  or  touches  the  upper  limit.  In  some 
partial  eclipses  where  this  curve  does  not  pass  near  the  zenith  of  the 
sphere  this  curve  will  not  reach  the  superior  limit. 

95.  The  vagueness  of  this  method  led  me  to  discard  it  wholly 
some  years  ago,  and  instead  I  make  use  of  Tables  XIII.  and  XIV., 
here  appended,  which  will  give  Q  sufficiently  exact  for  the  formulae. 
Table  XIII.  is  computed  by  formula  (165),  and  Table  XIV.  by 


95      N.   AND   S.    LIMITING   CURVES   OF   PENUMBRA.     109 

(166).  The  method  in  detail  is  to  first  lay  off  in  Fig.  8,  Plate  V., 
the  angle  E  from  the  centre  of  each  curve  ;  the  points  where  this  line 
crosses  the  outline  curves  are  marked  on  the  plan  by  a  single  dot ; 
sin  /9  is  the  distance  of  this  point  from  the  centre  of  the  sphere, 
which  is  to  be  carefully  measured  by  scale  for  each  point.  This 

distance  is  set  down  in  the  example  under  E,  Table  XIII.,  with  log  - 

€/ 

(which  is  constant  for  one  eclipse)  gives  the  angle  v.  With  this 
and  E,  Table  XIV.  gives  Q  the  first  approximation,  which  is  really 
the  second,  for  E  is,  in  fact,  the  first. 

This  value  of  Q  is  now  laid  off  by  a  protractor  for  each  curve  and 
marked  on  Fig.  8  by  two  dots.  Sin  /9  is  to  be  measured  again  for 
the  points  just  found,  and,  using  the  tables  in  the  same  manner,  we 
get  the  second  approximation  for  Q,  all  of  which  in  this  example 
are  so  near  the  previous  value  that  we  can  hardly  measure  any 
different  value  for  sin  /9,  so  we  may  take  the  last  values  of  Q  as 
final.  In  the  Wh  curve,  which  is  near  the  end  of  the  outline 
limits,  it  is  seen  that  Q  =  E,  nearly,  and  we  can  hardly  change  that 
value,  though  the  point  is  not  quite  at  the  end  of  the  eclipse.  If 
necessary,  Q  can  now  be  rigorously  computed  by  formulae  (165—166), 
on  which  the  tables  are  founded  ;  but  as  Q  is  not  needed  to  be  very 
exact,  as  CHAUVENET  explains  in  Art.  312,  this  will  not  be  neces- 
sary, especially  as  we  have  another  check. 

An  error  in  Q  will  shift  the  point  around  the  circle  of  shadow, 
and  approximately  along  the  limiting  line,  and  but  very  little  in  a 
lateral  direction,  so  that  an  error  in  Q  has  but  little  effect  in  moving 
the  position  of  this  curve. 

In  the  example  for  the  curve  we  are  obliged  to  compute  sin  /?  in 
order  to  get  cos  ft.  Sin  ft  in  numbers  to  three  decimals,  as  computed, 
is  0.442 ;  whereas  in  finding  Q  we  have  measured  for  this  curve  on 
the  plan  sin  ft  =  0.450,  which  the  succeeding  work,  by  its  agree- 
ment, proves  to  be  correct,  as  near  as  the  drawing  will  give. 

I  may  add  that  these  measurements  were  made  on  the  drawing  of 
which  Fig.  8  is  reduced  one-half.  But  in  drawings  of  the  same 
size  as  Fig.  8  as  good  results  may  generally  be  obtained  with  care 
and  accuracy  in  making  the  plot.  It  is  seen  from  the  example  that 
when  there  are  two  limiting  curves,  sin  ft  must  be  measured  for  each, 
and  §,  as  resulting,  does  not  differ  by  exactly  180°  for  the  two  points, 
since  Q  depends  upon  cos  ft,  which  is  different  for  the  Northern  and 
for  the  Southern  curves.  Sin  ft  must  be  measured  in  parts  of  radius  ; 
Fig.  8  being  drawn  to  a  scale  of  2J  inches  as  the  radius,  a  scale  of 


110  THEORY  OF  ECLIPSES.  95 

40  parts  to  the  inch  will  give  100  for  radius.  Table  XIII.  shows 
what  little  effect  log  -  or  e  has  upon  the  angle  Q  ;  which  is  the  rea- 
son it  may  be  taken  as  constant  in  this  curve.  E  and  Q  in  these 
tables  are  taken  as  acute,  being  laid  off  from  the  axis  of  Y  above  and 
below  the  centre ;  but  for  final  values  180°  is  added  for  the  Northern 
curve.  If  E  should  be  negative,  it  must  be  laid  off  toward  the  left 
hand,  which  is  the  negative  direction,  generally,  in  the  eclipse  theory. 

96.  EXAMPLE  OF  THE  USE  OF  TABLES  XIII.  AND  XIV. 

FOB  FINDING  Q — TOTAL  ECLIPSE,  1904,  SEPTEMBER  9. 

Northern  Curve.  Southern  Curve. 

Hour  Curve        8  9  10  8  9  10 

Q  =  E  16   12  17    35  18   58  16   12  17    35    18   58 

sin  /?  0.582  0.408  0.843  0.818  0.718      0.996 

Table  XIII.    v  20.1  22.2  13.5  14.5  17.3        2.4 

Table  XIV.     Q,  1st  Approx.  25.2  29.0  24.8  21.7  25.2        19.7 

sin/3  .515  .450  .887  .835  .690 

Table  XIII.    v  20.9  21.7  11.6  13.8  18.0 

Table  XIV.     Q,  2d  Approx.  25.7  28.6  23.6  21.2  25.6 

Values  205  42  208   36  203   36  21    12  25   36 

97.  The  Angles  Q  and  E. — The  angle   Q  being  found,  we  will 
examine  its  nature  and  use  and  relation  to  the  angle  E.     CHAUVE- 
NET  remarks  of  this  curve  that  "  it  is  commonly  regarded  as  one  of 
the  most  intricate  problems  in  the  whole  theory  of  eclipses,"  the 
reason  for  which  is  the  continually  changing  value  which  Q  takes. 
In  all  the  preceding  extreme  times  they  existed  but  for  the  instant 
of  contact — the  outline  also  for  the  instant  of  the  eclipse  hours. 
The  maximum  indeed  is  generated  by  the  motion  of  the  shadow ; 
but  Q  is  known,  since  it  equals  E  when  in  the  horizon.     But  here, 
while  the  shadow  moves  in  one  direction  over  the  fundamental  plane, 
the  surface  of  the  earth  is  also  moving,  but  in  quite  a  different  direc- 
tion ;  and  the  motion  of  the  shadow  on  the  earth's  surface  is  some- 
what akin  to  the  resultant  of  these  two  motions. 

We  may  illustrate  this  in  Fig.  15.  Let  M  be  the  centre  of  the 
shadow  moving  to  N  in  a  given  time — say  one  hour.  During  this 
time  the  surface  of  the  earth  at  M  has  moved  to  o,  or  p  has  moved 
to  M  and  q  to  N,  shortening  the  path  on  the  earth,  since  the  earth 
and  shadow  move  in  the  same  general  direction  ;  so  that  M  q  is  the 
direction  of  the  centre  of  the  shadow  on  the  earth's  surface.  If  the 
earth  were  stationary,  the  elements  of  the  cone  c  cf,  perpendicular  to 
M  N,  would  be  the  points  which  form  the  limiting  curves ;  but  by 


97      N.   AND   S.    LIMITING   CURVES   OF   PENUMBRA.     Ill 

the  two  motions,  e  e',  perpendicular  to  M q,  are  the  tangent  points, 
and  ef  is  the  southern  limiting  curve.  We  cannot  see  this  motion  in 
Fig.  8,  since  we  have  there  only  the  circles  centred  at  M  and  N  on 
the  fundamental  plane.  If  in  Fig.  13  we  draw  from  the  centre  N 

FIQ.  15. 


the  line  Ng  parallel  to  qf,  the  points  /  and  g  will  illustrate  the 
approximations  made  for  finding  Q  in  Fig.  8.  The  relative  dura- 
tion of  the  motions  in  Fig.  13  is  approximately  the  same  as  in  Fig. 
8 ;  for  example,  in  the  nine-hour  curve  the  earth's  surface  near 
the  southern  curve  would  move  in  an  ellipse,  which,  being  near  the 
axis  of  Yt  would  be  nearly  perpendicular  to  it ;  that  is,  diverging  on 
the  north  side  of  the  shadow  path,  as  in  Fig.  13,  Mo.  The  angle 
which  Me  makes  with  the  axis  of  Y  is  nearly  E,  and  the  angle  of 
Me  is  §,  the  angle  sought  for  the  limiting  curves.  This  will  illus- 
trate these  angles,  though  it  may  not  take  account  of  small  terms. 

98.  It  can  easily  be  shown  from  the  formulae  what  is  the  exact 
difference  between  the  angles  Q  and  E.  In  Article  300,  p.  463, 
CHAUVENET  gives  the  following  equation,  No.  (513) : 

P'  =  a'  +  e  sin  (Q  -  E)  -  C/sin  (Q  -  F)  (174) 

For  a  maximum  curve,  P'  =  0;  and  in  the  northern  and  southern 
limiting  curves  (Article  311),  the  simple  contact  is  the  maximum  of 
the  eclipse  at  that  point.  The  equation  is  simplified  by  omitting  the 
small  quantities  a'  and  F9  which  gives 

esm(<2-.E)=C/sm<2  (175) 


112  THEORY   OF   ECLIPSES.  98 

which  is  the  fundamental  equation  for  these  curves,  and  from  which 
equations  (161-66)  are  derived. 
Developing  the  above  e'quation, 

e  sin  Q  cos  E  —  e  cos  Q  sin  E  =  C/  sin  Q 
By  45, 


whence  tan  Q  =  -  -  -  =  -  —  —  (177) 

c'  —  C/      c'  —  f  cos  13 

And  by  176,  tan  E  =  ~  (178) 


c 


These  equations  show  the  difference  of  the  two  angles.  By  the  term 
£  entering  in  the  equation  for  Q,  it  is  seen  to  vary  with  the  height 
of  the  point  above  the  principal  plane,  and  when  f  =  0,  Q  =  E.  This 
term  being  subtracted  in  the  denominator,  it  follows  that  Q  >  E. 

99.  Approximate  Formulce.  —  These  do  not  vary  much  from  those 
already  given,  but  it  may  be  convenient  for  use  to  repeat  them. 

First  find  Q  by  Tables  XIII.  and  XIV.,  as  explained  in  Article 

95  and  example  Article  96.  (179) 

£  srs  sin  /3  sin  f  =  x  —  I  sin  Q  )  fl  8CH 

i)  =  sin  /?  cos  f  =  y  —  I  cos  Q  ) 

C=cos£  (181) 


c  cos  C  = 
cos  <p  sin  #  = 
cos  ?  cos^  =  c  cos  (<7+  d)  >  (183) 


(184) 


^ 

> 

J 


Compute  with  four-place  logarithms  to  minutes  for  every  thirty 
minutes  on  the  hours  and  half  hours.  This  curve  being  chiefly 
used  as  a  check  to  the  other  curves  on  the  chart,  is  the  least  impor- 
tant of  all  the  eclipse  curves,  and  no  important  observations  are 
made  near  it.  To  get  the  intermediate  thirty-minute  points,  arcs 
of  outline  curves  for  the  half  hours  can  be  drawn  in  pencil.  I  may 
be  taken  as  constant  for  the  whole  eclipse,  and  the  compression  is 
here  wholly  neglected,  so  that  e  =  0,  pl  =  1,  f  =  f,  <p  =  ft  ;  log  e  is 
constant  for  the  eclipse,  and  Table  XIII.  shows  what  small  effect  it 
has  for  the  small  changes  during  one  eclipse. 


100     N.   AND   S.   LIMITING   CURVES   OF   PENUMBRA.     113 

100.  Example. — Q  is  here  found  by  the  method  described  in 
Art.  95,  in  connection  with  Tables  XIII.  and  XIV.  In  these 
approximate  formulae,  since  e  is  omitted,  f  is  not  needed — only  its 
sine  or  cosine.  In  the  formulae  I3  -f-  rf  +  £2  =  1,  the  general  equa- 
tion of  a  sphere,  which  is  shown  by  the  example 


0.3522  +  0.2673'  +  0.897'  =  0.1239  +  0.714  -f  0.8046  =  0.9999. 

Now,  if  the  small  quantity  e,  equation  (168),  is  included,  this  equal- 
ity will  not  exist,  nor  will  it  exist  if  ^  is  substituted  for  17,  nor  if 
^>!  is  taken  for  <p.  It  is  necessary  to  recompute  the  coordinates  by 
the  group  given  under  the  Outline  Curves  Equations  (130-132). 
Moreover,  not  only  does  the  equality  above  not  exist,  but  sin  <p  and 
cos  <p  do  not  give  the  same  angle.  Such  great  accuracy  for  this  curve 
is  not  required,  and  the  only  alternative  is  to  omit  all  these  small 
quantities,  which  is  done  in  the  formulae  of  the  previous  section. 


NORTHERN  AND  SOUTHERN  LIMITING  CURVES. 
EXAMPLE— TOTAL  ECLIPSE,  1904,  SEPTEMBER  9. 


Tables 
Art.  96. 
(180) 


(180) 


9* 

I 

+  9.7264 

(181) 

Q 

208   36 

(182) 

sin  Q 

—  9.6801 

cos  Q 

—  9.9435 

XQ. 

+  8.9866 

I  sin 

—  9.4065 

I  —  IA 

0.4199 

B 

0.5599 

S 

+  9.5465 

(183) 

y<> 

—  9.3018 

(183) 

ICOB 

—  9.6699 

I—  IS 

0.3681 

A 

0.1252 

ij 

+  9.4270 

tan  7 

0.1195 

sin  y 

9.9011 

sin  /? 

-f  9.6454 

(184) 

Numbers 

0.442 

C 

+  9.9528 

tan  C 

+  9.4742 

C 

+  16   36 

cos  C 

9.9815 

logc 

+  9.9713 

<?  +  <£, 

+  21    52 

sin  ( 

)     +9.5711 

cos( 

)     +  9.9676 

ccos  ( 

)  +9.9389 

tan  $ 

9.6076 

$ 

+  22     3 

cos 

9.9670 

cos* 

9.9719 

sin* 

+  9.5424 

* 

+  20   24 

Pi 

135   42 

o 

+  113   39 

It  will  be  noticed  that  at  the  end  of  the  first  column  the  quantity 
sin  /?  is  computed,  resulting  0.442  in  numbers.  This  serves  as  a 
check  upon  the  approximate  value  0.450,  found  in  Art.  96,  which  is 
about  as  close  as  can  be  measured  on  a  drawing  of  the  size  used. 


114  THEORY   OF   ECLIPSES.  101 

101.   Curves  of  Any  Degree  of  Obscuration. — We  found   for  the 
Maximum  Curve  the  following  formula,  giving  the  magnitude 


A  being  the  distance  the  place  is  immersed  in  the  shadow,  if  we  sub- 
stitute J  for  I  in  the  formulae  of  article  99,  and  give  J  any  value 
between  0  and  unity,  the  equations  (179-184)  will  give  curves  north 
and  south  of  the  centre  line  on  which  the  eclipse  has  the  magnitude 
assumed  for  J.  As  L  and  L^  are  special  values  depending  upon 
the  elevation  of  the  surface  of  the  earth  above  the  fundamental 
plane,  /  and  ^  should  be  substituted  for  them,  and  the  previous 

equation  becomes 

A=l-M(l+l^  (185) 

And  from  (180) 

£  =  sin  £  sin  r  =  a;  —  J  sin  Q  )  ^^ 

y  =  sin  ft  cos  y  =  y  —  J  cos  Q, ) 

Then  proceed  with  the  equations  (181-184). 

The  error  of  using  I  here  instead  of  L  for  the  given  place  can 
never  be  as  much  as  0.01  of  the  sun's  disk  or  M  =  0.01,  and  usually 
will  be  much  less. 

If  M  be  assumed  0.1,  0.2,  0.3,  etc.,  the  series  of  curves  will  be 
approximately  parallel  to  the  centre  line,  the  two  values  of  Q  giving 
points  north  and  south  of  it.  The  value  0  for  J  would  give  the 
northern  and  southern  limiting  curves,  and  the  value  1.0  would 
give  the  northern  and  southern  limiting  curves  of  the  central  eclipse. 
All  these  lines  begin  and  end  on  the  Maximum  Curve  in  the  horizon. 

The  formulae  would  probably  have  to  be  computed  for  every  10 
minutes ;  and  I  can  be  taken  as  constant  for  the  whole  eclipse. 
These  curves  are  not  required  for  the  Nautical  Almanac,  nor  will 
they  often  be  required.  They,  however,  again  attest  the  thorough- 
ness of  CHATJVENET'S  treatment  of  this  subject. 

In  Harper's  Magazine,  vol.  iii.,  p.  239,  July,  1851,  is  a  diagram 
of  the  United  States,  showing  curves  of  obscuration  of  the  Solar 
Eclipse  of  July  28,  1851.  The  author  read  this  when  a  small  boy, 
was  interested  in  it,  and  his  memory  now  serves  him  to  recall  that 
it  illustrates  this  subject,  and  to  know  where  to  find  it. 


102          EXTREME  TIMES   OF   CENTRAL  ECLIPSE.          115 

SECTION    XL 
EXTKEME  TIMES  OF  CENTRAL  ECLIPSE. 

102.  Formula.  —  These  are  derived  by  CHAUVENET  from  those 
fgr  the  penumbra,  outline,  etc.,  by  the  consideration  that  the  radius 
of  the  shadow  is  zero,  that  is  (I—  i£)  —  0  ;  and  equation  (143),  which 
is  the  fundamental  formula  in  the  whole  theory  of  eclipse,  and  also 
the  following,  derived  from  it,  become  zero  : 

(I  —  t'C)  sin  Q  =  x  —  £  )  ri8?, 

(l-tC)ec»e«jf-?J 
whence 

x  =  £        and         y  =  i?  (188) 

For  the  extreme  times,  we  also  have  £x  =  0,  whence  by  equation  (142) 

£2  +  ri  =  1  (189) 

Also  by  the  above 

za  +  2/i2=l  (190) 

The  formulae  for  this  section  are  as  follows  : 

»-*  «•-£          "«> 

MO  sm  MQ  —  x0 


n  sn  -= 
ncosJV-y/ 

sin  ^  =  w0  sin  (Jf0  —  JV)  (194) 

T  =  cos^  __  m0  cos  (Ifp  —  N) 

n  .       7i 

T=T0+r  (196) 

r  =  N+</>  (197) 

cos  ^>i  sin  &  =  sin  ^  ^ 

cos  ^  cos  t?  =  —  cos  Y  sin  ^   >  (198) 

sin  <PI  =  cos  Y  cos  c^       J 


tan^=  (199) 

o»  =  ^  —  d  (200) 

103.  Compute  with  five-place  logarithms  to  seconds  of  arc.  Two 
approximations  are  necessary,  computing  first  for  the  epoch  hour. 
<p  has,  as  in  similar  equations,  two  values,  the  obtuse,  with  negative 
cosine  for  beginning  ;  and  the  acute,  with  positive  cosine  for  ending  ; 
MQ  and  log  m0  are  computed  but  once  for  the  epoch  hour.  For  the 
second  approximations  of  the  time  take  the  other  quantities  for  the 
times  just  found  ;  x0f  and  y/  are  the  mean  hourly  changes.  It  is 
generally  simpler  to  take  angles  which  result  from  these  formulae  as 


116  THEORY   OF   ECLIPSES.  103 

less  than  180°,  and  either  positive  or  negative  as  the  equations  denote. 
Angles,  however,  formed  by  addition  or  subtraction  may  result  greater 
than  180°.  The  second  approximation  of  the  times  will  generally 
vary  not  over  one  or  two-tenths  of  a  minute.  The  proportional 
parts  of  //!  can  be  taken  from  Table  VIII.  The  times  in  the  final 
approximation  should  be  gotten  to  four  or  five  decimals  of  a  minute, 
since  they  affect  the  longitudes. 

The  times  and  geographical  positions  of  the  central  eclipse  may 
also  be  computed  by  the  formulae  of  Section  V.  for  the  extreme 
times  generally,  by  placing  1  =  0. 

104.  Middle  of  the  Eclipse.  —  This  is  similar  to  preceding  cases. 


r=_i- 

n 
T=TQ  +  r  (202) 

105.  Check  Equations.  —  These  are  derived  from  equation  (189), 
and  are  similar  to  those  for  the  extreme  times  generally.     They  are 
computed  for  the  times  already  found. 

msinJ/=*)  (203 
ra  cos  M=  yi  ) 

Then  M  =  r  (204) 

m  =  unity.  (205) 

All  the  decimals  of  the  times  must  be  used  here  in  order  to  get  x 
and  7/j  sufficiently  exact  to  make  M  agree  with  f.  It  should  agree 
within  three  or  four  seconds  ;  m  should  be  within  one  or  two  units 
of  unity,  either  more  or  less.  Deviations  greater  than  these  may 
perhaps  not  affect  the  results.  Sine  ^  and  cos  ^  should  be  exam- 
ined to  see  if  they  give  the  same  angle  within  one  unit  of  the  last 
decimal.  These  check  equations  are  so  similar  to  those  in  Art.  57 
that  no  example  is  given. 

106.  Geometrical  Explanations.  —  The  points  found  by  the  above 
formulae  are  marked  K  and  L  on  Fig.  2,  Plate  L,  and  on  Fig.  3, 
Plate  II.,  giving  the  times  on  the  path  and  the  geographical  positions 
on  the  earth's  horizon  line.     The  formulae  can  be  explained  and 
shown  in  precisely  the  same  manner  as  in  Art.  58  for  the  extreme 
times  generally,  and  the  geographical  positions  as  in  Art.  60,  mak- 
ing some  few  changes  in  the  formulae  ;  for  example,  instead  of  p  -{-  I 
in  the  former  case,  we  here  have  p  =  m  =  />,  the  earth's  radius,  or 
=  unity,  as  these  differences  are  too  small  to  be  seen  on  a  drawing 
of  the  size  of  the  figures  here  given. 


106 


EXTREME   TIMES   OF   CENTRAL   ECLIPSE. 


117 


It  will  be  noticed  that  these  points  here  computed  are  not  points 
of  contact,  but  the  passage  of  the  centre  line  of  the  umbral  cone 
across  the  earth.  The  umbral  cone  of  course  forms  small  circles  of 
rising  and  setting  curves,  similar  to  those  of  the  penumbra,  and  sur- 
rounding the  points  K  and  L  on  the  earth's  surface ;  but  they  are 
never  computed.  The  points  K  and  L  lie  on  the  maximum  curve, 
already  computed. 

107.  Example. — The  first  approximation  in  one  column  is  here 
omitted,  since  the  first  is  precisely  like  the  second,  except  that  the 
latter  is  made  in  two  columns,  taking  out  the  quantities  from  the 
tables  for  the  times  given  at  the  head  of  the  columns,  which  are 
those  of  the  first  approximation.  No  quantities  in  the  first  approxi- 
mation are  used  in  the  second,  and  doubtless  the  reader  has  become 
so  familiar  with  the  methods  previously  adopted  that  he  will  require 
but  few  further  comments  as  a  guide. 

EXTREME  TIMES  OF  CENTRAL  ECLIPSE. 


EXAMPLE  —  TOTAL  ECLIPSE,  1904,  SEPTEMBER  9. 

From      first  \7  o  ftnn 
Appro*.  T  /  7  3'00 

10  25.728 

Nos.  (1) 

—1.68900 

+1.68896 

(192)  *0 

-f  8.98664 

-(2) 

—0.26073 

—0.26077 

y\ 

—9.30325 

T 

—1.94973 

+1.42819 

tan  M0 

9.68339 

(196)  T               f 

7.05027 

10.42819 

M0 

+154  14  54 

I 

7  3.0162 

10  25.6914 

sin 

9.60797 

9.37867 

(197)  y             +277  53  27 

+116  40  35 

cos 

9.95458 

(198)  sin  y 

—9.99587 

+9.95113 

Iogm0 

+9.34867 

cosy 

+9.13763 

—9.65220 

sin  d^ 

+8.96508 

+8.96081 

(193)  V 

+9.74646 

+9.74644 

cos  ^  cos  # 

—8.10271 

+8.61301 

<// 

—9.23934 

—9.23954 

tantf 

1.89316 

1.33812 

tanN 

0.50712 

0.50690 

&              —90  43  58 

+87  22  18 

N 

+107  16  49 

+107  17  18 

sin 

9.99996 

9.99954 

sin 

9.97994 

9.97993 

9.99591 

9.95159 

9.76652 

9.76651 

cos 

8.10680 

8.66142 

cos 

9.47283 

9.47302 

COS  0, 

+9.99591 

+9.95159 

log  1  :  n 

+0.23349 

+0.23348 

COSC?j 

+9.99814 

+9.99818 

(194)  M—  N 

+46  51  5 

+46  57  36 

sin0, 

+9.13577 

—9.65038 

sin(     ) 

+9.86390 

+9.86384 

tan^, 

+9.13986 

—9.69879 

cos(     ) 

+9.83404 

+9.83411 

(199)  tan  (j> 

+9.14133 

—9.70026 

sin  V 

+9.21257 

+9.21251 

0 

+7  53  0 

—26  38  0 

i> 

+170  26  38 

+9  22  17 

(195)  cos  V 

—9.99414 

+9.99414 

(200)  ^        106° 

26/22//   157°    7'  29 

log(l) 

—0.22763 

+0.22762 

«/      197 

10  20    +69   45  11  W. 

(2) 

+9.41620 

+9.41626 

I  —162 

49  40  E. 

(201)  Middle :  8ft.7393  =  8*  44"».358 


118  THEORY  OF   ECLIPSES.  107 

These  times  may  be  computed  by  the  formula  m  =p  in  the  same 
manner  as  shown  for  penumbra  in  Art.  57.  If  yl  is  used  for  m,  then 
the  time  is  when  m  =  unity  ;  but  if  y  is  used,  it  is  when  m  =  py  the 
earth  radius  for  the  place  of  beginning  or  ending. 


SECTION  XII. 

CURVE  OF  CENTEAL   ECLIPSE. 

* 

108.  Formulae.  —  Under  the  conditions  given  in  the  preceding  sec- 
tion, equation  (188),  the  formulae  for  outline  of  the  shadow,  Art.  81, 
reduce  to  the  following  simple  forms  (CHAUVENET,  Art.  315)  : 

y1  =  l  tf-*  (206) 

Pi  Pi 

sin  /?  sin  ?  =  x  j  ^^ 

sin/5  cos  r  =  yi) 

crin£=yi    0\  (208) 

c  cos  C  =  cos  0  J 

cos  ?>!  sin  #  =  x  1 

cos  ?!  cos  #  =  c  cos  (C  +  d^  V  (209) 

sin  <pl  =  c  sin  (  (7  -f  ^i)  ^ 
tan 


tan  <  — 


1/1-  e> 

<o  =  ^-  *  (211) 

#  =  The  Local  Apparent  Time. 

The  above  formulae  are  entirely  rigorous,  and  this  section  is  by 
far  the  most  important  of  the  whole  eclipse.  Compute  with  five- 
place  logarithms  to  seconds  for  each  ten  minutes  between  the  ex- 
treme times;  and  finally  interpolate  the  latitudes  and  longitudes 
for  every  five  minutes.  But  as  the  quantities  at  the  ends  vary  rap- 
idly, they  cannot  be  interpolated,  so  that  at  least  three  intermediate 
5-minute  points  must  be  computed  with  the  others  at  each  end  of 
the  curve. 
. 

109.  The  computer  should  first  interpolate  the  quantities  x,  y,  ply 
dl}  and  log  sin  dlf  in  the  Eclipse  Tables  for  the  5-minute  points.  As 
suggested  in  Art.  64,  if  log  x  and  log  y  have  already  been  gotten  for 


109  CURVE   OF   CENTRAL   ECLIPSE.  119 

the  Rising  and  Setting  Curve  to  five  places  of  logarithms,  the 
former  can  simply  be  copied  here,  and  log  yl  readily  gotten  from  y 
of  the  former  curve,  y  is  not  required  in  this  curve,  only  its  sine 
or  cosine.  As  this  curve  must  be  closely  computed,  either  sin  ft  or 
cos  ft  may  be  differenced.  The  difference  for  sin  ft  will  be  large  in 
the  middle ;  and  for  cos  ft  large  at  the  ends,  remembering  that  the 
end  points  are  computed  for  a  different  interval.  Another  very 
good  method  of  checking  quantities  in  these  curves,  where  the  dif- 
ferences at  the  ends  run  up  to  infinity,  is  to  subtract  one  end  from 
the  other.  These  differences  will  generally  be  much  less  than  dif- 
ferences in  the  usual  method  ;  and  this  series  can  now  be  differenced 
on  a  slip  of  paper  in  the  usual  manner.  The  addition  of  C+  d^ 
may  be  revised  if  necessary. 

Finally,  log  cos  ^  and  log  sin  <f>L  should  be  examined  to  see  that 
they  give  the  same  angle  within  one  or  two  units  of  the  last  decimal 
place  of  logarithms.  This,  as  previously  explained,  will  detect  any 
inconsistency,  but  not  any  other  kind  of  mistake. 

In  regard  to  differencing,  it  is  not  always  necessary,  but  it  some- 
times saves  much  time.  Small  errors  may  be  difficult  to  locate 
otherwise,  especially  toward  the  ends  of  the  curve,  or  if  two  or 
three  occur  near  together.  In  the  present  eclipse  there  are  twenty- 
six  computed  points  similar  to  that  given  in  the  example. 

Finally,  the  latitudes  and  longitudes  reduced  to  decimals  of  a 
minute  are  to  be  written  off  in  a  column,  differenced,  and  then  inter- 
polated for  every  five  minutes.  In  this  interpolation  for  one  or  two 
points  toward  the  ends  fourth  differences  are  necessary  to  make  the 
interpolation  difference  smoothly.  The  coefficient  for  this  is,  with 
sufficient  accuracy,  -^  of  2^,  and  even  this  will  not  always  make 
the  differences  run  smoothly. 

The  angle  &  in  these  formulae,  as  well  as  in  all  the  others,  when 
reduced  from  arc  to  time,  is  the  Local  Apparent  Time  of  the  phe- 
nomenon, a  very  useful  quantity  and  well  worth  a  place  in  the 
Nautical  Almanac  Tables  of  Eclipses.  It  may  be  reduced  to  Local 
Mean  Time  by  applying  the  equation  of  time  ;  and  is  therefore  valu- 
able to  astronomers  who  wish  to  observe  an  eclipse,  since  it  gives 
the  local  times  without  further  calculation. 

110.  Geometrical  Illustration. — These  formulae  are  essentially  the 
same  as  those  for  Outline  of  the  Shadow,  Section  VIII.,  after  the 
coordinates  £,  ??,  and  £  have  there  been  found.  In  the  present  sec- 
tion these  are  known,  or  at  least  readily  given  by  formula  (207) ; 


120 


THEORY  OF  ECLIPSES. 


110 


and  the  geometrical  explanation,  therefore,  does  not  differ  from  that 
already  given  in  Art.  85-6,  to  which  the  reader  is  referred.  The  points 
of  the  curve  on  the  earth's  sphere  are  those  on  the  centre  line  (Figs. 
2  and  8)  marked  by  the  10-minute  points.  The  angle  f  is  that 
which  a  line  drawn  from  any  one  of  these  points  to  the  centre  of 
the  sphere  makes  with  the  principal  meridian.  In  this  eclipse  f 
varies  from  —  82°  through  180°  to  +  116°. 

111.  Example. — In  the  following  example,  besides  the  computa- 
tion of  one  point  of  the  curve  of  central  eclipse,  there  is  given  also 
the  work  for  Duration  and  for  Northern  and  Southern  Limits. 
These  are  all  closely  connected,  and  the  two  latter  depending  upon 
various  quantities  in  the  computation  for  the  central  line ;  for  con- 
venience of  reference  they  are  therefore  given  together.  The  two 
last  columns  will  be  referred  to  under  their  proper  sections. 

CUKVE  OF  CENTRAL  ECLIPSE,  DURATION,  AND  NORTHERN  AND 
SOUTHERN  LIMITS  OF  UMBRA. 

EXAMPLE,  TOTAL  ECLIPSE,  1904,  SEPTEMBER,  9d  9*  0M. 


Central  Curve. 

Duration 

N.  and  S.  Limits. 

(207) 

X 

+8.98664 

(219)  log/, 

—8.1374 

(250)  L  :  cos  /3 

—8.2718 

(206-7)  yx 

—9.30325 

i  cos  (3 

+7.6536 

log  A 

—1.8081 

tany 

9.68339 

I  —  I  A 

0.4838 

sin£ 

+9.7042 

cos 

9.95457 

B 

0.6071 

(251)  cos  Q 

+9.9357 

sin/3 

9.34868 

L 

—8.2607 

sin  Q  sin  dl  +8.6668 

(208) 

cos/3 

+9.98891 

(220)  </ 

+9.7429 

tan  H 

+1.2689 

tan  C 

—9.31434 

/cos/3 

+9.4052 

H 

+86  55  6 

C 

—11  39  8 

I  —  IB 

0.3375 

sin 

+9.9994 

cos 

9.99096 

A 

0.0701 

log  h 

+9.9363 

logc 

+9.99795 

a 

+9.4753 

•&  —  H 

—81  19  9 

(209) 

(0+ 

dj    —6  23  16 

(221)  V 

+9.2438 

sin  (     ) 

—9.9950 

sin  ( 

)       —9.04627 

tan  Q 

0.2315 

cos  (     ) 

+9.1788 

cos  ( 

)       +9.99730 

sin  Q 

+9.9357 

(252)  log  d<f> 

+1.7394 

cos# 

costf*  +8.99525 

Numerator 

—2.0537 

log  (1) 

+9.9701 

tan# 

8.99139 

(222)  log  t 

—2.5784 

(2) 

—1.5105 

# 

+5  35  57 

t 

378.8 

Nos.  (1) 

+0.96 

cos 

9.99792 

Duration 

6"  18.8 

(2) 

—32.40 

COS0J 

+9.99733 

du 

—31.44 

sin^ 

—9.04422 

d*j>  . 

+54.88 

tan^ 

—9.04689 

(210) 

tan  (ft 

—9.04836 

Latitude.             Longitude. 

<S> 

—6  22  40 

^        (253)  N. 

Curve  —5°  27  '.8        129 

0  34'.3  W. 

/"i 

+135  41  42 

S. 

Curve    —7    17  .6        130 

37  .3  W. 

( 

+130    5  45 

(211) 

wt               130     5.8  W. 

[For  the  Duration,  see  Art.  117  ;  and  for  Limits,  Art.  138.] 

112  CENTRAL  ECLIPSE  AT  NOON.  121 

SECTION    XIII. 
CENTRAL  ECLIPSE  AT  NOON. 

112.  Formulae. — This  caption  means  at  Local  Apparent  Noon.     It 
is  the  point,  J,  where  the  centre  line  of  the  path  crosses  the  axis  of 
Y  (Plate  I.,  Fig.  2,  Plate  II.,  Fig.  3),  from  which  the  criterion  evi- 
dently is, 

*  =  0  (212) 

This  is  a  point  of  the  central  curve,  and  by  (209),  x  =  0 ;  also  by 
(206),  (207)  of  the  central  curve 

2/i  =  —  (213) 

Pi 

sin  J3  =  y,  (214) 

By  (208),  C  =  /?;  and  by  (209-211), 

K  =  ^  +  dj.  (215) 

tan  ^  =     tan  yi  ^  (216) 

w  =  ^  (217) 

The  Greenwich  mean  time  of  this  phenomenon  is  the  time  of  con- 
junction in  right  ascension,  already  found  for  the  elements,  Art.  21. 
Equations  (213-217)  solve  this  problem,  and  the  quantities  are  to- 
be  taken  for  the  time  of  conjunction.  If  the  precepts  of  the  fore- 
going sections  are  followed,  this  interpolation  will  be  for  a  fraction 
of  ten  minutes,  and  the  factor  in  numbers  from  the  example  (Art. 
21)  will  be  0.95686.  For  this  time  x,  interpolated  from  the  ten- 
minute  values  given  in  the  Eclipse  Tables  and  Nautical  Almanac, 
should  equal  zero,  and  not  vary  from  this  more  than  a  fraction  of 
the  last  decimal  of  numbers  or  logarithms. 

113.  The  following  equation  is  devised  as  a  check  upon  yl9  as  it 
gives  the  value  for  the  time  of  conjunction.      Since  x  =  0  here, 
(«  —  a)  =  0,  and  equation  (25)  for  y  becomes 

...        _         sin  (8  —  d) 
y  =  r  sm  (<5  —  d)  =  - 

sin  TT 

In  this  substitute  the  value  of  d  from  equation  (18). 

sin  [<j  _  sf  -| (d  —  <5')] 

y=-  _4^I& 

sin  TT 


122  THEORY   OF   ECLIPSES.  113 

which  is  rigorously  exact,  but  as  the  angles  are  small,  write  the 
angles  for  the  sign  and  reducing, 

y  =       *       -  (218) 

which  will  give  y  correctly,  using  five-place  logarithms.  If  d  —  df 
has  been  employed  in  the  elements,  as  suggested,  as  a  check  on  d 
and  3',  then  all  the  quantities  are  already  given.  This  equation  will 
be  again  referred  to  for  other  explanations  under  Section  XIX.,  on 
Prediction  of  Eclipses  by  Semidiameters,  Art.  159. 

114.  EXAMPLE,  CENTRAL  ECLIPSE  AT  NOON. 

EXAMPLE,  TOTAL  ECLIPSE,  1904,  SEPTEMBER  9. 

(218)  From  the  Elements  (Art.  21)            6  —  6'  —MX  25".61  —2.79631 

TT  +61   22  .957  +3.56619 

—  9.23012 

From  Main  Computation  (Art.  36)  log  1 : 1  —  6  -f  0.00103 

y                         -0.170272  —9.23115 

As  interpolated  as  a  check  y  —  0.170276 

(214)  (ft  from  tables  =  9.99854)                  y,  —9.23261 

p  —  90  50'  14" 

<*!  -f-5    16     2 

(215)  tan^/S  +  di  —  4    3412        —8.90271 

(216)  tantf*     0  —4    35     7        —8.90417 

(217)  From  Eclipse  Tables  (Art.  37)          o  =  //i  -f  133      5  11  West. 

The  latitude  and  longitude  from  the  central  line  should  now  be 
interpolated  for  the  time  of  conjunction,  and  they  should  agree  with 
those  just  found.  The  interpolation  is  for  a  fraction  of  five  minutes. 
The  time  of  conjunction  in  the  present  example  being  (Art.  21)  8A 
49"*.57  is  4m.57  after  the  five-minute  time,  and  the  fraction  for  inter- 
polation is  +  0.914;  second  differences  should  be  used. 

115.  Special  Cases. — In  the  eclipse  1891,  June  6,  y  at  conjunction, 
as  we  see  from  the  eclipse  tables  for  that  year,  has  the  value  about 
0.9962  in  numbers ;  log  y,  9.9983,  and  log  yl  about  9.9945,  which 
gives  rather  uncertain  values  for  /?,  a  fault  of  the  formula  which 
cannot  be  avoided,  being  near  90°.  ft  is  about  87°  20',  c?x  about 
-f  22°  40' ;  consequently,  p  +  d,  =  110°,  and  ^  =  70°.  The  shadow 
having  fallen  beyond  the  north  pole,  we  must  therefore  take 

V  =  180°  —  (/?  +  dj  =    70°  north 

to  =  K  +  180°     =  250°  west  =  110°  east. 


115    DURATION   OF   TOTAL  OR  ANNULAR   ECLIPSE.    123 

This  is  the  eclipse  noted  in  Art.  67  as  being  an  Annular  Eclipse 
of  the  Midnight  Sun. 

Impossible  Solution. — In  the  eclipse  of  1896,  Feb.  13,  y  at  con- 
junction has  a  value  of  about  —  1.02,  yv  will  be  still  greater,  and 
the  equation  sin  ft  =  yl  will  show  an  impossible  value  for  /?.  The 
centre  line  does  not  cross  the  principal  meridian  of  the  earth's 
sphere,  and  there  is  consequently  no  Eclipse  at  Noon.  This  eclipse 
is  also  referred  to  in  Art.  67.  The  preceding  case  approaches  very 
nearly  to  this  :  y  when  divided  by  pl  is  not  far  from  unity. 

The  quantity  y  or  yl  at  conjunction  is  the  most  convenient  element 
for  measuring  the  motion  of  the  shadow  north  or  south  in  successive 
eclipses  of  a  series,  since  the  inclination  of  the  path  changes  .with 
each  eclipse.  The  latitude  at  Noon  cannot  be  used,  since  the  motion 
is  rapid  at  the  poles  and  slow  toward  the  zenith  of  the  projection. 
The  durations  cannot  be  used,  for  their  differences  are  zero  toward 
the  equator  and  rapid  at  the  poles. 


SECTION    XIY. 

DURATION  OF  TOTAL  OR  ANNULAR  ECLIPSE. 
116.  Formulas  (CHAUVEKET,  Art.  317)  : 

L^lt  —  h  cos  {3  (219) 

a  =  c'  —  /  cos  /3  (220) 

tan§  =  ^  (221) 

7200  L  sin  Q 


This  computation  can  conveniently  be  placed  below  that  of  the 
central  line,  on  account  of  quantities  in  the  former  work  required 
also  here  being  computed  for  the  same  times  as  the  central  line. 
All  the  quantities  are  to  be  taken  for  the  umbral  cone.  Compute 
with  four-place  logarithms.  Regard  the  sign  of  llt  and  in  fact  all 
the  signs,  though  t  is  to  be  taken  as  positive,  even  if  the  equation 
gives  a  negative  value.  The  logarithm  of  bf  can  generally  be  inter- 
polated with  care,  unless  it  is  very  small  ;  in  which  case  the  natural 
numbers  must  be  interpolated  and  the  logarithms  gotten  from  them. 
Log  cf  can  always  be  easily  interpolated  here,  and,  as  suggested  in 
Art.  32,  c'  is  computed  for  the  umbral  cone.  Addition  and  subtrac- 
tion logarithms  will  be  found  very  convenient  for  these  formulae. 


124  THEORY   OF   ECLIPSES.  116 

The  above  quantities  are  given  on  a  basis  of  one  hour's  interval  ; 
especially  br  and  c',  which  are  formed  from  the  hourly  motions  of  x 
and  y.  t  would  therefore  result  in  decimals  of  an  hour,  and  the  con- 
stant 3600  is  introduced  to  give  t  in  seconds.  And  as  the  duration 
depends  upon  the  diameter  of  the  shadow,  whereas  we  have  used  /, 
the  radius,  the  factor  2  is  introduced  ;  t  may  result,  a  negative  value 
from  L  being  negative  for  total  eclipse,  but  it  must  be  taken  as 
positive  in  all  cases,  since  it  is  the  numerical  interval  of  time  that 
is  desired. 

Q  is  not  required  here,  only  its  sine.  But  as  the  cosine  is  required 
for  the  limiting  curves,  it  saves  time  to  take  out  both  from  the  tables, 
and  set  down  the  cosine  ten  or  twelve  lines  below  for  future  use. 

117.  Example.  —  This  will  be  found  in  Art.  Ill,  near  the  compu- 
tations for  the  Central  Line.     Cos  f)  is  to  be  taken  from  that  com- 
putation.    It  need  not  be  copied  off,  for  i  cos  /?  and  also  /  cos  /? 
can  be  written  off  at  once,  having  log  i  and  log  /  on  a  slip  of  paper 
with  other  constants  for  the  eclipse.     The  computation  need  be  car- 
ried no  further  than  log  t,  getting  t  in  seconds  and  reducing  to  min- 
utes while  transferring  to  another  page,  where  it  is  differenced  for 
errors,  and  then  interpolated  for  every  five  minutes.     One  decimal 
of  a  second  only  is  sufficient.     This  series  of  total  eclipses  has  the 
longest  duration  of  any  others,  except  the  series  of  1883,  1901,  etc., 
which  is  increasing,  while  the  duration  of  the  series  of  1886,  1904 
is  decreasing. 

118.  The  Angle  Q.—  By  comparing  equations  (220),  (221),  of  this 
section  with  equation  (177)  of  the  Limiting  Curves  of  Penumbra,  it 
is  seen  that  the  numerators  and  denominators  of  tan  Q  are  trans- 
posed in  the  two  forms,  showing  apparently  that  the  two  values  are 
complements  of  one  another.     The  truth  is,  however,  that  Q  in  this 
section  is  Q  of  the  previous  section  plus  90°,  which  is  easily  shown 
by  CHAUVENET'S  formulae  in  vol.  i.,  p.  493.     The  equations  are  not 
numbered,  but  are  as  follows,  substituting  L  in  the  first  members  : 


sn 


x'  —  £'  =  c'  —  /  cos  /5  =  a 

y'  -y'=-V 


118  EXTREME  TIMES—  N.  AND  S.  LIMITS  OF  UMBRA.   125 

In  substituting  the  latter  equations  for  the  former,  CHATJVENET 
omits  the  signs,  "  since  it  is  only  the  numerical  value  of  t  that  is 
required  "  ;  but  if  we  retain  the  signs  we  can  see  the  value  of  Q  as 
follows  —  the  upper  signs  are  for  the  point  of  beginning  of  the  dura- 
tion, and  the  lower  for  the  point  of  ending  : 


(223) 


Taking  the  lower  signs,  Q  is  an  obtuse  angle  ;  and  with  the  upper, 
an  acute  angle  taken  negatively  —  the  values  differing  180°  from  one 
another.  In  Fig.  15,  supposing  the  circles  to  now  represent  the 
umbral  shadow,  while  we  plot  for  the  northern  limiting  curve  Q  — 
A  Me'  ,  we  have  here  for  duration  for  the  point  of  beginning  Q  = 
AMm  =  BnMj  which  is  laid  off  in  the  direction  of  the  path,  the 
points  m  and  n  forming  the  duration.  A  total  eclipse  would  reverse 
the  conditions  of  the  signs,  on  account  of  the  negative  value  of  jL. 

We  read  in  the  section  on  Limits  that  the  path  of  the  shadow  on 
the  earth's  surface  is  shortened  from  the  distance  on  the  fundamental 
plane,  and  the  office  of  Q  is  here  to  diminish  the  radius  of  the  umbral 
cone  proportionately,  so  that  instead  of  the  duration  depending  upon 
the  radius  n  M  =  Mm,  it  depends  upon  the  line  ms,  which  is  propor- 
tionate to  Jfm,  as  sin  Q  is  to  unity.  It  is  as  well  to  know  these  facts 
about  the  angle  §,  but  they  are  given  here  not  as  a  matter  of  curios- 
ity, but  because  this  knowledge  will  shorten  the  labor  of  computing 
the  umbral  limiting  curves  by  nearly  one-half. 

As  Q  in  the  former  sections  hardly,  if  ever,  reaches  35°,  and  is 
generally  much  less,  its  value  here  when  the  signs  are  neglected  will 
in  effect  be  between  +  40°  and  135°. 


SECTION   XV. 

EXTREME  TIMES,  NORTHERN  AND  SOUTHERN  LIMITS  OF  UMBRA. 
119.  CHAUVENET'S  formulas,  (Art.  320) : 

m  sin  M  =  x0  zp  h  sin  E0  )  /^oo/<\ 

__   r  \LL'±) 

m  cos  M  =  y0^p  h  cos  ^  ) 

(225) 

n  cos  N  =yQ'  ^f.  —  cf 
e 


cos 
cos 


126  THEORY   OF   ECLIPSES.  119 

The  above  are  CHATJVENET'S  formulae,  but  the  results  will  be 
more  accurate  if  yl  and  yj  be  substituted  for  y0  and  y0',  as  sug- 
gested in  Art,  130  and  Art.  134.  Then  ?  =  f. 

(  For  Total  Eclipse:  —for  north  limit,  +  for  south  limit.  ) 
I  For  Annular  Eclipse  reverse  these  conditions.  ) 

sin  (p  =  m  sin  (M  —  N)  (227) 

T  =  cosjA  _  mcaa(M  —  N) 

n  n 

T=T0  +  r  (229) 

r  =  N+</>  (230) 

tan  f  =  Pi  tan  r  (231) 

<f>i  sin  #  =  sin  ?'  ~\ 

<pl  cos  #  =  —  cos  r'  sin  c^  C  (232) 

sin  <f>l  =  cos  Yr  cos  e?!      3 

tan  <pi  /ooo\ 

tan  ^  =  _  (zoo) 

1/1  —  e* 

«  =  A»I  -  «  (234) 

{^  is  obtuse  for  beginning  with  its  cosine  negative  ; 
And  acute  for  ending  "          "       positive. 

120.  These  formulas  are  to  be  computed  closely  with  five-place 
logarithms,  because  figures  are  given  in  the  Nautical  Almanac,  and 
also  because  they  are  to  be  compared  with  the  results  for  the  central 
line. 

It  will  be  noticed  that  although  the  quantities  x0,  yoy  EQ  are  taken 
for  the  epoch  hour,  yet  the  formula?  reduce  them  to  the  time  of  the 
phenomenon.  This  is  readily  seen  from  CHAUVENET'S  construction 
of  the  formulae  on  the  upper  half  of  page  486,  vol.  i.  It  is  also  seen 
in  formula  (225),  because  we  have  from  equation  (45),  Art.  32,  for  Ey 

bf  cf 

sin  E  =  —  cos  E  =  — 

e  e 

The  mean  hourly  changes  of  XQ  and  yQ  are  xa'  and  y0'  ;  they  are  to  be 
taken  for  the  times  of  the  central  eclipse.  The  hourly  changes  of  bf 
and  c'  are  60r/  and  c0r/  ;  hence,  the  hourly  changes  of  l±  sin  E0  are 

therefore  -Lb"  ',  and  of  ^  cos^0,  J_  c0";  so  that  equation  (225)  is  made 

6  C> 

up  wholly  of  the  hourly  changes  of  equation  (224). 

The  quantity  log  e,  as  shown  in  Art.  44,  has  a  minimum  value  at 
the  middle  of  the  eclipse,  and  its  variation  before  and  after  this  time 


120  EXTREME  TIMES— N.  AND  S.  LIMITS  OF  UMBRA.   127 

is  symmetrical  for  equal  times.  Hence,  for  the  times  of  beginning 
and  ending  of  the  eclipse,  its  values  are  found  to  be  sensibly  equal. 
That  is  the  reason  why  CHAUVENET  in  the  example  (p.  487)  takes  e 
as  constant  and  at  the  ends  of  the  eclipse. 

But  one  other  quantity  remains  to  be  considered,  the  radius  of 
umbra  lr  The  formulae  take  no  account  of  its  changes,  and  CHAUVE- 
NET in  his  example  takes  a  mean  value,  and  employs  only  four-place 
logarithms.  This  is  not  up  to  the  standard  of  accuracy  of  the  com- 
putations heretofore  given  where  figures  are  required.  We  should 
take  ^  for  both  the  times  of  beginning  and  ending  of  the  central 
eclipse,  giving  two  values  of  M  and  log  m ;  and  all  the  data  will 
then  result  for  the  times  of  beginning  and  ending. 

121.  These   formulae   are   the   same   precisely  as   given    for  the 
penumbra,  using,  however,  1L  for  umbra  instead  of  I.     CHAUVENET 
gives  very  meagre  directions  for  their  use  for  the  umbral  times,  and 
omits  wholly  the  important  fact  that  when  used  for  a  total  eclipse 
the  negative  sign  of  ^  would  reverse  the  conditions  of  the  signs  given 
for  penumbra.     The  proper  signs  to  be  taken  are  noted  here. 

On  account  of  the  several  conditions  imposed  upon  these  formulae — 
the  number  of  columns,  the  exactness  required  in  the  use  of  five-place 
logarithms — these  formulae  will  probably  try  the  computer  more  than 
any  other  part  of  the  whole  eclipse.  The  results  coming  out  so  near 
together,  and  having  to  agree  with  those  of  the  central  eclipse,  small 
mistakes  become  more  apparent  than  in  the  previous  problems. 

As  the  times,  latitudes,  and  longitudes  here  lie  close  to  those  of 
the  extreme  times  of  central  eclipse,  the  several  quantities  may  be 
taken  for  those  times  without  error,  so  that  but  one  approximation  is 
sufficient.  It  is  suggested  that  the  computer  take  the  four  columns 
in  the  order  given  in  the  example,  and  also  mark  the  headings  as 
given  there  with  the  signs  to  be  used.  Addition  and  subtraction  loga- 
rithms are  almost  indispensable  here.  Throughout  this  computation 
the  several  quantities  will  result,  so  that  the  corresponding  quantity 
for  central  eclipse  will  usually  lie  between  them,  the  exceptions  being 
rare.  It  will  be  noticed  that  the  angle  f  is  used  in  these  equations, 
while  f  is  used  in  the  central.  The  proportional  parts  for  //D  as 
before  noted,  can  be  taken  from  Table  VIII. 

122.  Example. — A  part  of  the  work   is  the   same  for  the  two 
points  for  beginning  and  for  the  two  points  for  ending,  so  that  it 
need  not  be  computed  for  both.    The  double  sign  then  gives  the  four 


128 


THEORY   OF   ECLIPSES. 


122 


points.     As  suggested  in  Art.  22,  the  computer  is  supposed  to  have 

the  constants  log  — j  log  — =   =>  etc.,  written  on  a  slip  of  paper, 
e  1/1 —  e2 

since  they  are  not  written  in  the  example.  In  these  equations  there 
is  some  similarity  of  numerical  values  that  are  generally  the  same  in 
all  eclipses.  In  the  portion  depending  upon  the  equations  (224),  (225) 
(see  example),  the  values  of  the  two  northern  points  are  generally 
alike,  and  also  those  for  the  two  southern  points ;  for  example, 
m  sin  M.  But  the  angle  N  is  alike  for  the  two  points  for  begin- 
ning, and  a  different  value  for  the  two  points  for  ending.  The  num- 
bers (1),  (2)  are  used  for  the  two  terms  of  equation  (228),  and  for 
the  second  term  the  two  beginnings  are  generally  similar,  and  the 
two  endings.  The  latitudes  should  be  checked  by  ascertaining  if 
sin  (p^  and  cos  <pL  give  the  same  angle  within  at  least  one  unit  of  the 
last  place  of  logarithms.  The  differences  of  /^  for  the  two  points 
of  beginning  and  for  ending  can  be  checked  by  the  proportional  parts 
for  the  small  interval  of  time  between  them. 


EXTREME  TIMES  N.  AND  S.  LIMITS  OF  UMBRA. 
EXAMPLE,  TOTAL  ECLIPSE,  1904,  SEPTEMBER  9. 


(124) 


(224,  226) 


(224) 


(224) 


(225) 


log/, 
sin  EQ 


Beginning. 

N.  (-).  S.  (+). 

—  8.13894 
+  9.48029 
+  9.97920 


Ending. 

N.  (-).  S.  (+). 

—  8.13821 
+  9.48029 
+  9.97920 


x0  +  8.98664 

^sin^    —7.61923 
I  —  I          41.36741 
51.31566 
msinM    +9.00489 

j/o  —  9.30179 

^cosEo    —8.11814 
I  —  I          51.18365 
A  1.15423 
mcosJlf    —9.27237 


tanJf  9.73252            9.63823            9.73265            9.63838 

M  +151°  37'  26"  +156°  307 12"  +151°  37'  0"  +156°  29'  46" 

cos  9.94441            9.96241            9.94938            9.96239 

logm  +9.32796       +9.36694       +9.32803       +9.36692 


+  8.98664 

—  7.61850 

5 

A  1.36814 

5 

A  1.34835 

51.38656 

A  1.34913 

+  8.96758 

+  9.00506 

+  8.96769 

—  9.30179 

—  8.11741 

A 

5  1.18438 

A 

51.21121 

41.15500 

51.21190 

—  9.32935 

—  9.27241 

—  9.32931 

log(lre)  +0.2362 
6"  +8.1268 

c"  —7.6182 


+  0.2362 
+  8.1268 
—  7.6182 


122  EXTREME  TIMES— N.  AND  S.  LIMITS  OF  UMBRA.   129 


(225,  226) 

V 

+  9.74646 

+  9.74644 

(J,»«)ft» 

—  6.5019 

—  6.5012 

J  —  / 

^4  3.2446 

B 

A  3.2452 

B 

(B-A) 

B    25 

A     25 

B    25 

A    25 

n  sin  .AT 

+  9.74671 

+  9.74621 

+  9.74669 

+  9.74619 

(225) 

y/ 

—  9.23788 

—  9.23808 

(/!  :  e)c" 

+  5.9933 

+  5.9926 

/  —  *• 

.43.2346 

B 

A  3.2455 

B 

(B-A) 

B    24 

A    24 

B    24 

A    24 

n  cos  N 

—  9.23812 

—  9.23764 

—  9.23832 

—  9.23784 

(225) 


tanN  0.50859  0.50857  0.50837  0.50835 

N         +107°  13'  31"  +107°  13'  33"  +107°  14'  0"  +107°  14/  3" 
sin  9.98007  9.98007  9.98005  9.98005 

log(l:n)  +0.23336        +0.23386       +0.23336       +0.23386 


(227) 

M—N  +44°  23'  55" 
sin  (     )     +  9.84488 
cos  (     )     +  9.85400 
sinV>    '     +9.18284 

+44°  16'  39" 
9.87960 
9.81451 
9.24654 

+44°  23'  0" 
+  9.84476 
+  9.85411 
+  9.17279 

41°  15'  43" 
+  9.87950 
+  9.81465 
+  9.24642 

V»          +171°  26'  16"  +169°  50'  20" 

+8°  33'  40" 

+10°  9'  30" 

(228)  Beginning—  \ 
En  ding  +       / 

cosV 

—  9.99513 

—  9.99314 

+  9.99514 

+  9.99314 

log  (1) 

—  0.22849 

—  0.22700 

+  0.22850 

+  0.22700 

(2) 

+  9.41532 

+  9.41531 

+  9.41550 

+  9.41543 

First  terra 

Nos.  (1) 

—  1.69235 

—  1.68654 

+  1.69238 

+  1.68654 

Second  term 

-(2) 

—  0.26020 

—  0.26020 

—  0.26036 

—  0.26027 

r 

—  1.95255 

—  1.94674 

+  1  .43202 

+  1.42627 

(229) 

T 

f  7.04745 

7.05326 

10.43202 

10.42627 

\       ^y 

1  7*  2.8470 

7*  3.1956 

10*  25.9212 

10*  25.5762 

(230) 
(231) 
(232) 


y  +278°  39'  47"  +277°  3'  53"  +115°  47'  40"  +117°  23'  33" 

tany  0.81712            0.90682            0.31579            0.28552 

tany'  0.81566            0.90536            0.31433            0.28406 

yf  +278°  41'  31"   277°  5'  18"  +115°  52'  12"  +117°  28' 15" 

sin  y'  —  9.99498 

cos  Y  +  9.17932 

sin  ^  +  8.96508 

cos  0,  sin  £  —  8.14440 

tan  tf  1.85058 

£  —90°  48'  29"  —90°  39X  27"  +87°  27r  47"  +87°  16r  49" 

sin  9.99996     9.99997     9.99957     9.99951 


—  9.99667 

+  9.95414 

9.94804 

+  9.09131 

-  9.63981 

9.66398 

+  8.96508 

8.96081 

8.96081 

—  8.05639 

8.60062 

8.62479 

1.94028 

1.35352 

1.32325 

<233) 

(234) 
<234) 


COS^ 

+  9.99502 

+  9.99670 

9.95457 

9.94853 

cosrf, 

+  9.99814 

9.99814 

9.99818 

9.99818 

sin  0! 

+  9.17746 

9.08945 

9.63799 

9.66216 

tan  0j 

+  9.18244 

9.09275 

9.68342 

9.71363 

tan^ 

+  9.18391 

9.09422 

9.68489 

9.71510 

0 

+8°  41'  0" 

+7°  4'  53" 

—25°  49'  25"  —27°  25'  33" 

106°  23'  49"      106°29/3"      157°10/55"     157°5X44" 

f  +1 97°  12'  18"  +197°   8X  30"  +69°  43'   8"  +69°  48'  55" 
1—162  47  42    —162  51  30 


130  THEORY  OF   ECLIPSES.  123 

123.  The  above  results  may  now  be  compared  with  those  of  central 
eclipse  in  the  following  manner  : 

Times.  <£.  #.  «. 

N.  Limit  begins  7*   2™85  ,   iy     +8°  41/.0_48  Q  —90°  48'.49  ,  4  51  —162°  47/.7_2  Q 
Central       "73  ^T  7  53  •0_48'1      90  43  -QST^      162  49  .7_1'8 


S.  Limit     "73  .20  +7     4  .9  —90  39  .45  —162  51  .5 

S.  Limit  ends  10  25  .58  ,  n  +27  25  .5_47  5  +87  16  .82  ,  g  4g  +69  48  .9_3  ? 
Central  "  10  25  .69^T'23  26  38  -0_48'6  87  22  .SOjV^g  69  45  .2_9'1 
N.  Limit  "  10  25  .92  +25  49  .4  >  +87  27  .78  +69  43.1 

124.  And  to  show  the  changes  during  successive  periods  of  the 
Saros,  the  Nautical  Almanacs  furnish  the  data  : 

Durations.  At  Noon.  Extreme  Dura- 

Eclipse.  Extreme.  Central.  <£.  tion  of  Totality. 

1868,  Aug.  17,  5*  14*6  3*  25-3  +10°  27'.2 

1886,  Aug.  28,  5    14  .0  3   24  .2  +2    58  .5  6m  33«.6 

1904,  Sept.  9,  5   12  .9  3   22  .7  —  4    35  .1  -      6    23  .7 

125.  These  quantities  difference  very  much  more  smoothly  than 
those   of  any  other   eclipse  I  can  remember.      In  CHAUVENET'S 
example  the  times  of  the  limits  for  beginning  differ  from  the  cen- 
tral times  by  the  quantities  Om.96  and  Om.54,  and  the  corresponding 
longitudes  differ  by  43'  and  35'.     After  computing  these  times,  I 
found  a  small  mistake  affecting  the  four  columns,  which'  changes 
some  of  the  results  a  few  tenths  in  the  decimal  ;  but  it  was  too  late 
to  make  the  change  in  the  Almanac.     The  errata,  however,  appear 
in  the  Almanac  for  1906. 

These  differences  are  generally  so  irregular  that  the  comparison 
with  the  central  times  cannot  be  relied  upon  at  all  as  a  check  upon 
their  accuracy.  I  have  noticed  that  a  large  eclipse  like  the  present, 
where  the  shadow  path  approaches  the  zenith  point  of  the  projection, 
will  difference  more  smoothly  than  one  in  which  the  centre  of  shadow 
passes  near  the  poles. 

126.  It  will  be  noticed  by  reference  to  Fig.  8,  Plate  V.,  that  the 
line  a  6,  Fig.  8,  which  makes  the  angle  E  with  the  meridian,  and  is 
tangent  to  the  sphere,  is  not  quite  at  right  angles  to  the  centre  line 
of  the  path.     If  the  inclination  of  the  path  were  such  that  this  line 
should  be  at,  or  nearly  at,  right  angles  to  the  path,  then  the  central 
eclipse  would  begin  before  either  of  the  limiting  curves  ;   and  the 
central  times,  instead  of  lying  between  them,  as  in  the  comparison 
above  given,  would  lie  outside  of  them  —  a  rather  unusual  occur- 
rence ;  but  it  happened  in  the  eclipse  of  1894,  Sept.  28.     The  longi- 


126  EXTREME  TIMES— N.  AND  S.  LIMITS  OF  UMBRA.   131 


tudes  are  published  in  the  Almanac,  but  not  the  times ;  they  both 
show  this  peculiarity,  and  are  as  follows  : 

S.  curve  ends,  1894,  Sept.  28,  19*  14W.29.      Longitude,  -  162°  41'.2 
Central      "  19   14  .11  —  162    43  .5 

N.  curve    "  19   14  .48  —  162    36  .5 

Analogous  to  this  in  the  present  eclipse,  it  is  seen,  in  Art.  92,  that 
the  extreme  times  of  central  eclipse  lie  outside  of  the  limits  of  the 
penumbra,  which  is  also  unusual. 

127.  Check  Formula?. — There  is  no  rigorous  check  that  I  know  of 
to  these  times  and  geographical  positions.  Some  years  ago,  however, 
I  devised  a  very  simple  formula  which  will  check  the  differences 
between  the  quantities,  though  it  cannot  check  the  quantities  sepa- 
rately. 

Navigators  will  doubtless  recognize  these  formula?  as  a  transforma- 
tion of  those  used  in  Middle  Latitude  Sailing,  and  they  are  easily  de- 
rived. As  the  shadow  in 

the   present  eclipse  first  FIG.  16. 

strikes  the  earth  for  the 
southern  limiting  curve 
at  the  point  a  (Fig.  16, 
A),  the  parallels  of  lati- 
tude of  a  and  b,  with 
their  hour  angles,  will 
form  a  quadrilateral  on 
the  earth's  surface  (Fig. 
16,  I>),  of  which  ab  is  the 

diagonal.  The  earth  revolves  a  little  before  the  first  point  of  the  nor- 
thern limiting  curve  is  formed,  which  is  the  point  c  of  Fig.  16,  B.  In 
this  latter  figure  db  =  ae  is  the  difference  of  the  hour  angles  Ad  ;  ad 
the  difference  of  latitude,  J<p  ;  ab  is  the  arc  of  a  great  circle  =  m 
of  the  formula?  and  in  the  horizon  of  the  first  figure,  be  is  the  differ- 
ence between  the  quantities  fa  for  these  two  points,  and  dc  their 
difference  of  longitude.  The  line  a  c  cannot  be  geometrically  shown, 
and  it  is  seen  that  the  difference  of  hour  angles,  and  not  of  longitudes, 
must  be  used  in  the  formulae.  They  are  as  follows  : 


m  sn 
m  cosM 


A<p 
m—  Ay 


(235) 
(236) 


132 


THEORY   OF   ECLIPSES. 


127 


Cos  <p  here  is  the  mean  of  the  two  latitudes  of  the  points  a  and  b. 
In  fact,  the  quantities  are  so  small  it  makes  little  difference  whether 
the  latitude  of  either  a  or  6  or  the  mean  is  used,  or  if  the  geocentric 
latitudes  are  substituted.  As  J^>  and  A&  are  taken  in  minutes,  m 
will  also  result  in  minutes. 

Four-place  logarithms  are  amply  sufficient  for  these  formulae.  It 
will  be  noticed  that  while  7-  is  used  for  the  central  curve,  f,  not  f,  is 
used  for  the  limiting  curves,  and  this  value  must  be  used  in  the 
formulae  above.  Signs  may  be  wholly  neglected  here,  considering 
only  numerical  differences.  All  the  quantities  required  are  given  in 
the  computations  (Articles  107  and  122).  The  times  are  checked  by 
these  formulae  only  indirectly,  and  by  inference — that  if  these  quanti- 
ties check  correctly,  we  may  assume  that  the  times  are  correct. 

128.  EXAMPLE  OF  THE  CHECK  FORMULAE. 


CHECK  ON  THE  EXTREME  TIMES,  LIMITS  OF  UMBRA. 
Between  Central  Point  and  Limits. 


Beginning. 


Ending. 


(135)  (Art.  123)  Atf  4.51 

"        cos  ^ 

m  sin  M 

(Art.  123)  mcoslf=A^  48.0 
tan3/ 
cosM 


N. 

0.6542 
9.9954 
0.6496 
1.6812 
8.9684 
9.9981 


4.53 


48.1 


s. 

0.6561 
9.9963 
0.6524 
1.6821 
8.9703 
9.9981 


5.48 


48.3 


N. 

0.7388 
9.9528 
0.6916 
1.6839 
9.0077 
9.9978 


5.48 


47.5 


0.7388 
9.9498 
0.6886 
1.6767 
9.0119 
9.9977 


48.20  1.6831    48.31  1.6840    48.54  1.6861    47.75  1.6790 


(136) 


Check 


Angles  of  Position. 
N.      7'278°41'.51_4806 
Cent,  7    277   53  -45_48  15 
S.       7'  277     5  .30 

N.  /  115  52  .20+48  39 
Cent,  y  116  40  .59J_47  66 
S.  Y  117  28  .25^ 


48.06 
+  0.14 


48.15 
+  0.16 


48.39 
+  0.15 


47.66 
+  0.09 


Between  the  Two  Limiting  Points. 


(135)  (Art.  123)  Atf 
cos  $ 
m  sin  M 
(Art.  123)  m  cos  M  1 


Beginning. 
9.04    0.9562 
9.9959 
0.9521 


Ending. 
10.96    1.0398 
9.9513 
0.9911 


tan  M 
cos  M 


(136) 


Check 


96.1    1.9827      95.8    1.9814 

8.9694  9.0097 

9.9981  9.9977 

1  36.52  1.9846  1  36.32  1.9837 
1  36.21  1  36.05 

+  0.31  +  0.27 


In  the  above  example  the  formula  is  first  applied  to  the  extreme 
points  of  the  central  curve  and  limits,  and  below  that  to  the  points 
of  the  two  limiting  curves. 


128  EXTREME  TIMES— N.  AND  S.  LIMITS  OF  UMBRA.   133 

The  check  errors  are  not  in  the  check  formulae,  but  in  CHATJVE- 
NET'S  approximations;  because  the  former,  while  theoretically  ap- 
proximate, are  nearly  rigorous  as  applied  to  such  small  quantities  as 
here  used.  And,  moreover,  the  following  rigorous  formulae  give 
results  agreeing  very  closely  with  the  results  of  the  check  formulae : 

cos  m  =  sin  y>1  sin  ^2  +  cos  <?i  cos  ^2  cos  (^$)  (237) 

sm 

The  first  of  these  requires  seven-place  logarithms,  and  is  entirely 
rigorous,  ^  and  <p2  being  the  latitudes  of  the  two  points  previously 
used;  but  the  formula  has  the  disadvantage  of  giving  a  small  angle 
by  its  cosine.  If,  however,  the  angle  is  uncertain,  the  second  formula 
may  then  be  used,  employing  only  four-  or  five-place  logarithms ;  the 
sin  m  is  to  be  found  from  the  cosine  of  the  first  formula  and  dividing 
by  sin  1';  m  results  in  minutes  and  the  formula  is  accurate  to  a  small 
fraction  of  a  second. 

129.  Errors  in  CHAUVENET'S  Formulas. — It  is  to  be  regretted  that 
Professor  CHAUVENET  passed  over  this  subject  so  briefly  in  a  single 
page,  merely  referring  to  his  previous  formulae  for  the  penumbra  which 
are  to  be  used  here.  By  his  using  only  four-place  logarithms  in  the 
example  when  the  formulae  will  bear  five  or  more,  and  taking  lly  an 
approximate  value  for  the  middle  of  the  eclipse,  we  are  led  to  ques- 
tion whether  he  was  entirely  satisfied  with  the  formulae. 

That  the  reader  may  see  at  a  glance  what  errors  in  the  formulae  are 
suspected,  the  following  comparisons  of  the  times  of  ending  are  given 
for  the  eclipse  of  1903,  Sept.  20.  This  eclipse  is  selected  because 
the  variations  are  large  and  the  geographical  positions  are  in  high 
latitudes. 

TOTAL  ECLIPSE,  SEPTEMBER  20,  1903. 

CENTRAL  ECLIPSE  AND  LIMITS  OF  UMBRA  AT  ENDING. 

From  the  Nautical  Almanac : 

<*>.  w. 

N.  Limit  ends  — 80°  48'.2_,  -„-  —179°    8'.2  ,35g 

Central       "  82     0  .3_Q  2?1  178    32  .3  Jo, ' 

S.  Limit     "  —82    27  .4  —178    10  .7"1"' 

From  the  computing  sheets  : 

*•  y  y. 

N.  Limit  ends  17ft  27-.S4    „  , ,       -f-S2°  43'.20_,  7  ,q  yf  170°  5CK.92  ,, 

Central       «     17    25  .73      '  81    35  .67     *  ''if  y   172     3  .92 j"! 

S.  Limit     «•     17   25  .12-°'61      +81     5  .00~°30'67  \>  172   31  .37+° 


134  THEORY   OF   ECLIPSES.  129 

These  quantities  have  been  checked  by  the  formula  in  Art.  127, 
and  are  correct  within  0'.2,  so  that  there  is  no  error  of  the  computa- 
tion to  be  considered.  The  irregularities  before  mentioned  are  seen 
in  all  the  quantities,  especially  in  the  latitudes.  The  centre  line 
being  in  the  middle  between  the  two  limits,  it  is  not  reasonable  to 
suppose  there  should  be  such  unequal  differences  as  shown  above. 

An  examination  for  these  irregularities  leads  to  the  discovery  also 
of  two  other  sources  of  error  in  the  formulae,  for  all  of  which  the 
corrections  are  quickly  and  easily  made. 

130.  First. — An  error  in  applying  the  formulae ;  that  is,  in  taking 
no  account  of  the  change  in  E  during  the  intervals  between  the 
beginnings  or  endings  of  the  northern  and  southern  curves,  when 
the  error  is  sufficiently  great  to  have  any  appreciable  effect.     This  is 
well  illustrated  in  the  annular  eclipse  of  July  18,  1898,  from  which 
we  have  as  follows  : 

Change  of  E  for  one  hour,  6°  16'.9,  or  for  one  minute,       6'.28 
Interval  between  times  of  beginning  of  N.  and  S.  limits,     6™.  7 
Change  of  E  in  this  interval,  42'  5" 

This,  however,  is  rather  an  extreme  case.  No  account  of  any  such 
change  is  provided  for  in  the  formulae,  because,  first,  the  times  of 
limits  are  unknown,  and  consequently  their  interval  from  the  cen- 
tral ;  and,  secondly,  because  the  quantities  #',  yn ',  /T  are  taken  for 
the  times  of  central  eclipse,  and  the  computation  made  for  those 
times.  The  above  is  the  total  change  between  the  limits ;  in  practice 
it  would  be  computed  for  each  of  the  four  points  and  applied  to  EQ 
in  the  formula  (224).  The  correction  is  large  only  during  the  sum- 
mer and  winter  solstices,  when  sin  d  is  numerically  large ;  and  also 
when  the  path  is  much  inclined  to  the  axis  of  X,  both  of  which  aug- 
ment the  quantity  6',  equation  (40).  The  intervals  of  time  are  large 
when  the  centre  line  strikes  the  earth  very  obliquely.  This  correction 
would  require  two  approximations  of  the  times. 

131.  The  second  error  is  small,  affecting  the  beginnings  alike  and 
the  endings  alike,  but  with  contrary  signs.   In  CHAUVENET'S  approxi- 
mation of  his  equation  (530),  page  480,  vol.  i.,  or  equation  (174)  of 
the  present  work,  he  places  the  small  quantities  equal  to  0 ;  and  this 
constitutes  the  second  error.     If  they  are  retained,  we  have 

.;.       '  .in(g-.E)--n'  +  ygin<g-F)          .        (239) 


131   EXTREME  TIMES— N.  AND  S.  LIMITS  OF  UMBRA.   135 

When  this  is  used  for  limiting  curves,  £  has  a  value  approaching 
unity,  and  the  second  term  becomes  more  correct  toward  the  middle 
of  the  eclipse  and  a!  becomes  equal  to  0,  so  that  CHAUVENET'S 
approximations  are  correct,  especially  as  it  is  shown  in  Art.  95  that 
the  error  will  be  along  the  curve,  and  not  laterally. 

But  when  this  formula  is  applied  to  the  extreme  times,  it  is  not  a 
line,  but  a  point,  which  is  required  ;  a'  is  a  maximum,  and  the  error 
becomes  apparent.  For  these  times,  the  points  being  in  the  funda- 
mental plane,  £  =  0,  and  we  have  rigorously 

sin(§  —  E)=  —  —  (240) 

e 

For  a',  equation  (48) 

a'=—l'  —  /V  i  x  cos  d  (241) 

This  last  formula  shows  that  the  correction  varies  with  x  ;  it  is  0  in 
the  middle  and  has  contrary  signs  at  the  ends. 

The  above  corrections  for  the  total  eclipse,  Sept.  9,  1904.  x  = 
unity,  or  very  nearly,  at  the  beginning  and  end. 

Beginning.  Ending. 

ft  i,  x  cos  d  —  0.001205  +  0.001205 

£/  +  51  37 

log  (—a')  —  7.06221  +  7.06744 

logl:e  -f  0.23620  +  0.23617 

log  sin  ( Q  —  E )  =  —  7.29841  +  7.24361 

Q  —  E=      —  0°  6'  50"  +  0°  V  55" 

This  is  easily  computed,  for  the  quantities  are  all  given  in  the 
computation,  and  JEj,  for  the  epoch  hour  corrected  by  these  results 
will  give  their  effect  to  the  beginning  and  ending. 

This  correction  varies  with  x  more  than  with  any  other  factor,  so 
that  it  may  be  near  its  maximum  for  this  eclipse. 

132.  The  third  error  is  the  most  important  of  all,  for  it  is  perhaps 
the  whole  source  of  the  irregularity  noticed  in  the  tabulated  quanti- 
ties of  Art.  129.  The  error  is  that  yt  and  its  hourly  motion  y/ 
should  be  used  in  the  formulae  instead  of  y  and  yr.  CHAUVENET'S 
formulae  give  the  times,  using  y  and  y'  without  taking  any  account  of 
the  compression  of  the  earth.  He  then  considers  the  compression  by 
reducing  f  to  f  for  the  latitudes  and  longitudes ;  but  this  is  not 
enough :  the  compression  affects  the  times  and  the  quantities  M,  m,  N9 
n,  and  through  these  </>,  and  finally  f.  The  times  affect  ft  and  the 
longitude. 
t  CHAUVENET,  in  discussing  the  extreme  times  of  the  limits  of 


136 


THEORY   OF   ECLIPSES. 


132 


penumbra,  had  occasion  to  refer  to  his  equation  (514),  which  is 
£2  H~~  y\  =  1>  *n  which  he  substitutes  37,  and  naturally  uses  y  instead 
of  y1  in  the  equations  following.  The  times  are  unimportant,  and 
used  merely  as  a  check  on  positions  computed  only  for  the  chart 
in  the  Nautical  Almanac,  so  that  the  error  is  not  perceived  here  and 
the  equations  are  sufficiently  correct  for  the  penumbra.  But  when 
he  gives  the  formulae  for  the  umbral  cone,  he  briefly  refers  to  these 
for  penumbra  with  the  remark  that  ^  for  umbra  should  be  used 
instead  of  I. 

The  proper  quantities  to  be  used  here  are  yl  and  y/.  They  are 
used  for  the  extreme  times  of  central,  the  central  curve,  outline  from 
which  all  the  other  formulae  are  in  a  measure  derived.  These  quanti- 
ties are  not  used  for  the  unimportant  problems  of  rising  and  setting 
curves,  maximum,  the  limiting  times  and  curves  of  penumbra,  and 
finally  for  the  extreme  times  generally.  In  this  latter  case  he  has 
made  use  of  a  simpler  device  than  employing  fa  to  take  account  of 
the  compression.  He  finds  the  radius  of  the  earth,  jo,  for  the  point 
of  first  and  last  contacts,  and  this  quantity  also  enters  into  the 
formulae  for  <j)  and  f. 

133.  To  show  the  effect  of  using  y±  and  y/  upon  the  quantities 
given  in  Art.  129,  these  have  been  recomputed,  with  the  following 
results ;  the  quantities  on  the  lower  lines  can  be  compared  with  those 
previously  given : 

COMPARISON  OF  EXTREME  TIMES,  LIMITS  OF  UMBRA,  AT  ENDING. 
TOTAL,  ECLIPSE,  1903,  SEPTEMBER  20. 


M 

Using  y  and  y'. 
N.  Curve.       S.  Curve. 
+  185°  47'  40"   185°  57'  47" 

Using  ?/i  and  y-{  . 
N.  Curve.       S.  Curve. 
185°  46'  29"   185°  56'  35" 

logm 

9.95941 

9.96539 

9.96087       9.96684 

N 

107°  51'  55" 

107°  52'  22" 

107°  55'  21"   107°  55'  56" 

log  1  :  n 

0.24527 

0.24542 

0.24515       0.24528 

$ 

62°  57'  10" 

64°  37'  30" 

63°  18'  0"   64°  59'  40" 

Also  the  following  results,  using  y\'  and  y\r : 


N.  Limit  ends 
Central       " 
S.  Limit     " 

N.  Limit  ends  17A27"M1 


—81°  10'.4 
82      0  .3 


—82    51.2 


Central 


17   25  .73 


—  lm.38 


S.  Limit     "      17   24  .31 


— 49'.9 
—50  .9 


24/>42 
81    35  .67 

+80    34  .57 


—179°    0'.6 

178    32  .3 

—177    52  .6 


+39.7 


—  0  48.75 


-171°  13'. 

172      3.92 
—172    55.60 


133  CURVE—  N.  AND   S.  LIMITS   OF   UMBRA.  137 

Quantities  from  the  central  curve  are  unchanged.  The  above  results 
are  checked  by  the  formulae  of  Art.  127  and  are  correct  within  0'.2. 
They  seem  to  be  much  more  in  accordance  with  what  we  might 
reasonably  expect  than  those  in  Art.  129  given  by  CHAUVENET'S 
formulae.  The  differences  between  the  values  for  central  eclipse  and 
the  two  limits,  it  seems,  should  be  nearly  if  not  exactly  equal. 

134.   Corrected  Formulas.  —  For  the  convenience  of  the  computer  the 
changes  above  suggested  in  the  formulae  of  this  section  are  as  follows  : 

Art.  131,  0'  =  —  //  —  w&  cos  d  (242) 


Art.  131,  sin  (ft  -«)-__  (24g) 

e  e 

Art.  130,  Q=Q0  +  MEl  (244) 

Art.  132,  7,!=^-  y/=^  (245) 

Pi 


,    , 

n  cos  N 

t  is  the  interval  of  time  of  the  limits  after  the  central. 
AE  is  the  change  of  E  in  one  unit  of  the  time  t. 

The  rest  of  the  formulae  of  Art.  119  remains  unchanged  except  that 
f  needs  no  reduction  to  f. 

It  will  be  noticed  that  on  account  of  the  corrections  in  equation 
(243),  it  is  no  longer  E  that  is  used  in  these  formulae,  but  Q.  The 
correction,  equation  (244),  requires  two  approximations  of  the  times, 
the  first  being  used  to  obtain  the  interval  t. 


SECTION  XVI. 

CUEVE,  NORTHERN  AND  SOUTHERN  LIMITS  OF  UMBRA. 

135.  Remarks. — In  this  curve  f  or  cos  ft  is  not  known,  but  as  the 
curve  lies  very  near  central  line,  cos  /?  is  taken  from  that  curve,  and 
the  formulse  are,  therefore,  not  rigorously  exact,  but  are  sufficiently 
so  for  ordinary  purposes,  and  the  error  becomes  less  as  cos  /?  becomes 
greater — that  is,  toward  the  middle  of  the  eclipse  where  greater 
accuracy  is  desired. 


138  THEORY   OF   ECLIPSES.  135 

As  £  or  cos  ft  enters  as  a  divisor  in  the  equation  for  A,  near  the 
ends  where  it  is  small  X  becomes  very  large  and  at  the  extreme 
points  becomes  infinite.  Nevertheless,  I  have  computed  points  of 
this  curve  one  minute  after  the  commencement  of  the  central  curve 
with  good  results  for  the  limiting  curves.  These  errors  of  the  for- 
mulae are  of  little  account,  for  no  important  observations  are  made 
at  the  ends  of  the  eclipse,  because  the  sun  is  low  in  the  horizon  and 
much  affected  by  errors  of  refraction  and  parallax. 

136.  Formulas  (CHAUVENET,  Art.  320): 

For  Q.  tan  v  =  ^cos  /?  (248) 

€ 

tan  (Q  —  £  E)  =  tan  (45°  +  v)  tan  £  E  (249) 

Curve  A  =  I  — ^   — 

\cos  ft  sin  1' 

A-T7  (250) 

cos/S  sin  1' 

AsiniZ"=cos§  ) 

h  cos  H—  sin  Q  sin  dl  j 
d<p  =  tii  sin  (#  —  jff) 
dot  =  tii  cos  (#  —  H)  tan  ^  -f  A  sin  Q  cos 
For  Total  Eclipse  North  Curve  is         <p  +  dtp     to  -j-  dot  ^| 

South  Curve  is         <f>  —  dy     at  —  d<i>  V  (253) 

For  Annular  Eclipse  reverse  these  conditions.  J 


137.  The  Angle  Q. — The  formulae  for  Q  are  the  same  as  those  used 
for  the  penumbral  curves  (Arts.  93  and  98).  And  by  comparing 
the  formula  (177)  for  limits  of  penumbra,  with  (221)  for  duration  of 
central  eclipse,  it  is  seen  that  Q  for  limits  of  penumbra  is  the  com- 
plement of  Q  for  duration  ;  consequently,  the  value  of  Q  need  not  be 
computed  for  this  curve  if  the  duration  has  already  been  computed. 
For  tan  Q  is  there  given,  from  which  its  sine  is  used  in  that  curve ; 
and  it  is  suggested  at  the  end  of  Art.  116  that  the  cosine  be  also 
taken  out  of  the  tables  at  the  same  time  and  written  ten  or  twelve 
lines  below  for  use  in  this  computation.  It  should  here  be  marked 
sine;  and  the  sine  in  duration  can  now  be  copied  from  the  work, 
placed  below  the  sine  here  and  marked  cosine.  This  suggestion 
reduces  the  computation  of  this  curve  nearly  one-half. 

An  example  of  the  computation  of  Q  is  here  given  for  the  eclipse 
of  Sept.  9,  1904.  The  cosine  agrees  exactly  with  that  used  in.  the 
duration,  Art.  Ill,  but  the  sine  differs  two  units  of  the  last  decimal. 


137  CURVE— K  AND   S.  LIMITS   OF   UMBRA.  139 

To  make  them  agree  exactly,  either  one  or  both  would  have  to  be 
carried  out  to  seconds,  or  at  least  to  tenths  of  a  minute  of  arc. 

For  these  formulae  two  values  of  Q  are  given,  differing  180°  ;  but 
as  the  numerical  values  depending  upon  them  are  the  same  with  dif- 
ferent signs,  only  the  acute  value  of  Q  is  to  be  taken,  and  the  change 
of  signs  is  taken  account  of  in  equation  (253),  above  given. 

EXAMPLE,  COMPUTATION  FOR  §,  LIMITS  OF  UMBRA. 

(249)  E  +17°  36' 

(1  :  2)  #  8     48 


T.  9".  (K 

From  Central  Curve  cos  (3  +  9.9889 

From  Eclipse  Table  /  +  9.4163 

"          "  "       log(lre)  +0.2365 

(248)  tan  v'  +  9.6417 

v'  +23   40 

tan  (45°  +  v')  +  0.4083 


tan  9.1898 

tan  [Q—  (1:2)£]  9.5981 
Q—(l:2)E  +21  37 
Q  +30  25 

sin  Q  +9.7044 

cos  Q  +9.9357 


138.  Example.— This  will  be  found  in  Art.  Ill  with  that  of  the 
central  curve.  Compute  with  four-place  logarithms  to  seconds  or  a 
fraction  of  a  minute  for  the  same  times  as  the  central  curve ;  and 
this  computation  can  conveniently  be  placed  below  that  for  duration, 
since  quantities  from  both  of  these  computations  are  required  here. 
E  log  e  llt  etc.,  must  be  taken  as  variable  for  the  times  heading  the 
columns  for  central  curve.  Cos  ft  tan  <ply  &  are  to  be  taken  from  the 
central  curve,  ^  or  L  from  the  duration  ;  also  sin  Q  and  cos  Q  as 
above  noted,  or  else  computed  for  this  curve.  The  signs  for  these 
are  the  same  as  those  for  E. 

For  X  the  first  form  of  equation  (250)  is  CHAUVENET'S,  but  the 
second  form  will  be  found  more  convenient,  taking  L  from  duration, 
having  regard  to  its  sign,  and  cos  /9  from  central  without  copying 
either;  then  with  1  -*-  log  sin  V  (3.5363),  A  is  at  once  gotten  and  may 
be  conveniently  placed  above  sin  Q.  Sin  dt  is  interpolated  for  10 
minutes  in  the  Eclipse  Tables,  and  need  not  be  copied  here,  but 
used  directly  from  the  tables.  H  is  an  angle  always  near  +  90°  and 
varying  slowly.  &  —  H  varies  between  the  ends  of  this  curve  about 
160°  from  near  180°  through  270°  to  near  0°. 

d<p  and  dco  need  not  be  written  here  as  in  the  example,  but  carried 
over  to  another  page,  placed  in  columns,  and  differenced  for  errors, 
and  interpolated  for  every  five  minutes.  They  can  then  be  applied 
to  the  latitudes  and  longitudes  of  the  central  line,  giving  its  limiting 
curves. 

These  quantities  vary  so  rapidly  at  the  ends  that  five-minute  points 
must  be  computed  as  directed  for  the  central  curve,  and  even  then  for 


140  THEORY   OF   ECLIPSES.  138 

the  succeeding  one  or  two  intermediate  points  fourth  differences  must 
be  used,  for  which  the  coefficient  is  +  -^j-2'J4. 

139.  Peculiarities  of  these  Curves. — In  some  eclipses  the  values  of 
d<p  and  dco  become  very  large  at  the  ends.     In  the  total  eclipse  of 
Aug.  8,  1896,  dco  is  over  5°,  and  the  line  joining  the  northern  and 
southern  limiting  points  makes  a  very  acute  angle  with  the  central 
line.   And  generally  the  line  across  the  path  whose  extremities  gener- 
ate the  limiting  curves  is  far  from  being  a  normal  to  the  path.    More- 
over, dtp  sometimes  changes  the  sign  for  one  or  two  points  at  the  very 
ends.    This  is  also  the  case  in  the  above-mentioned  eclipse,  in  which 
the  peculiarity  is  seen  (Plate  IV.,  Fig.  7,  H)  that  the  latitude  of  the 
southern  point  is  greater  than  that  of  the  corresponding  northern 
point  of  the  limiting  curves.    As  published  in  the  Nautical  Almanac, 
the  latitudes  and  longitudes  stand  thus : 

G.  M.  T.  Northern  Limit.  Central  Line.  Southern  Limit. 

Limits.  +63°  10'.6     0°  57'.4  W.  +62°  51'.5    0°20'.5W.  +62°  18'. 2    0°  49'.2  E. 

15*  55  67    20  .0   12     6  .9  E.  67    46  .1  17    23  .6  E.  68    12  .2  21    46  .3 

16     0  71    24  .3   34    53  .6  71      3 .8  38    31  .8  70   43  .3  42    10 .0 

16     5  4-72   43  .4  52   16  .0  +72     6  .2  54   55  .9  +71    29  .0  57   35  .8 

This  eclipse  is  reproduced  from  the  chart  of  the  Nautical  Almanac, 
in  Fig.  18,  Plate  VII.,  wherein  this  peculiarity  is  seen  to  be  correct. 
As  this  eclipse  is  of  the  same  series  as  CHATJVENET'S  example,  the 
same  peculiarity  is  seen  on  his  page  (503),  the  latitude  of  the  southern 
point  of  the  limiting  curve  at  1A  0™  is  50°  57',  and  greater  than  that 
of  the  northern  curve,  which  is  given  50'  18.  This  looks  very  much 
like  a  mistake,  but  it  is  correct. 

At  the  beginning  and  end  of  the  eclipse  the  earth's  surface  is 
inclined  at  a  great  angle,  nearly  90°  to  the  fundamental  plane,  and 
consequently  at  a  very  small  angle  to  the  cone  of  total  shadow,  so 
that  for  a  very  small  change  of  the  angle  Q  round  the  circle  of 
shadow,  the  points  will  be  moved,  one  west,  the  other  east,  at  great 
distances  geographically  on  the  earth's  surface.  This  is  the  explana- 
tion of  the  increased  values  of  these  quantities  at  the  ends  of  the 
eclipse. 

140.  Another  peculiarity  sometimes  seen  the  computer  may  meet 
with  :  For  the  point  nearest  to  the  extreme  points  there  will  some- 
times be  a  point  to  one  curve  and  none  to  the  other.     In  the  eclipse 
of  Feb.  22,  1887,  the  north  curve  begins  at  7h  55m.9,  central  at  7A 


140  THE   CHART.  141 

58W.7,  south  curve  at  Sh  lm.l.  A  point  of  the  central  curve  is  com- 
puted at  8A  0'".0,  with  the  durations  and  limits,  but  as  the  northern 
extreme  point  begins  before  this  time,  there  can  be  no  point  of  the 
curve,  for  it  has  not  yet  commenced.  The  geographical  position  can 
of  course  be  plotted,  but  it  will  be  found  to  lie  on  the  other  side  of 
the  maximum  curve.  At  the  end  of  this  eclipse  the  same  thing 
occurs  :  the  south  curve  has  ended  before  the  central,  and  there  is  no 
point  to  the  north  curve.  For  these  points  a  blank  is  left  in  the 
Almanac  in  the  column  of  geographical  positions.  I  remember 
having  handed  in  several  eclipse  pages  showing  this  peculiarity, 
but  the  line  has  evidently  been  struck  out  before  printing,  and  the 
peculiarity  thereby  lost. 

This  completes  the  computations  for  the  eclipse.  The  chart,  how- 
ever, is  yet  to  be  made  and  the  geographical  positions  plotted  thereon, 
for  which  the  reader  is  referred  to  the  next  section. 


SECTION    XVII. 

THE  CHAET. 

141.  THE  curves  and  geographical  positions  computed  by  the 
foregoing  formulae  are  now  to  be  plotted  upon  a  chart,  showing  the 
whole  eclipse  at  a  glance.  Generally,  and  except  in  unusual  cases, 
the  eclipse  will  resemble  either  one  of  the  two  charts  here  repro- 
duced :  Fig.  17,  Plate  VI.,  from  the  original  drawings  made  by  the 
author  for  the  Nautical  Almanac,  and  Fig.  18,  Plate  VII.,  from  the 
chart  in  the  Nautical  Almanac,  as  the  original  drawing  could  not  be 
found.  The  former  is  the  total  eclipse  of  1904,  Sept.  9,  taken  as  an 
example  throughout  this  work.  It  shows  the  two  ovals  forming 
detached  branches  of  the  rising  and  setting  curves,  connected  by 
two  limiting  curves  of  the  penumbra. 

The  second  figure  is  the  total  eclipse  of  1896,  Aug.  8,  of  the  same 
series  as  that  of  CHAUVENET'S  example,  the  eclipse  of  July  19, 
1860,  and  selected  for  that  reason.  In  this  figure  the  rising  and 
setting  curve  is  continuous — in  shape,  that  of  a  distorted  figure  8, 
the  ends  connected  by  one  limiting  curve  of  penumbra.  This  is  the 
most  common  form  of  eclipse,  the  previous  form  occurring  only  in 
very  large  eclipses. 


142  THEOKY  OF  ECLIPSES.  141 

In  partial  eclipses  the  shape  is  like  Fig.  18,  but  the  central  line  or 
path  is  wanting.  They  dwindle  down  to  special  forms  mentioned  in 
Art.  67  in  very  small  eclipses. 

142.  Kinds  of  Projection. — There  are  three  which  may  be  avail- 
able, the  Globular,  Equidistant,  and  Stereoyraphic. 

143.  The  Globular  Projection. — All  points  of  the  sphere  are  pro- 
jected obliquely  by  converging  lines,  piercing  the  primitive  circle 
and  meeting  in  a  point  in  the  axis  of  that  circle,  at  a  distance  below 
the  lower  surface  of  the  sphere  equal  to  the  square  root  of  2.     The 
condition  is,  that  points  of  the  sphere  of  45°  elevation  shall  be  pro- 
jected exactly  half-way  between  the  primitive  circle  and  its  centre, 
and  the  converging  point' results  as  above.     This  projection  distorts 
the  surface  of  the  sphere  toward  the  borders  of  the  drawing. 

144.  The  equidistant  projection  is  a  modification  of  the  above.  When 
the  pole  is  the  primitive  circle,  the  semidiameters  representing  the 
equator  and  principal  meridian  are  divided  into  nine  equal  parts ; 
also  the  four  quadrants  are  likewise  so  divided,  and  circles  are  drawn 
through  these  points,  three  of  which  determine  the  circle.     These 
form  the  meridians  and  parallels  of  latitude.     When  the  pole  is  ele- 
vated, a  slight  modification  of  this  appears  necessary.    This,  like  the 
former  projection,  distorts  the  surface  of  the  earth  as  the  circles  do 
not  all  meet  at  right  angles,  especially  near  the  corners  of  the  draw- 
ing.    This  is  probably  the  projection  made  use  of  by  the  English 
Nautical  Almanac. 

145.  The  Stereogra.phic  Projection. — This  is  also  made  in  the  fun- 
damental plane  by  converging  lines  which  meet  in  the  axis  of  the 
primitive  circle  and  at  the  lower  surface  of  the  sphere.*     This  pro- 
jection contains  a  number  of  important  properties,  the  two  princi- 
pal of  which  are  that  all  circles  of  the  sphere  project  into  circles, 
and  also  that  the  parallels  of  latitude  and  meridians  intersect  one 
another  at  right  angles.     On  this  account  there  is  but  little  distor- 
tion ;  but  parts  are  magnified  toward  the  borders  of  the  drawing. 

*  Professor  CHARLES  DA  VIES,  in  several  of  his  excellent  works  on  geometry  and 
projections,  is  mistaken  in  saying  that  the  eye  is  supposed  to  be  at  this  point  on  the 
lower  surface  of  the  sphere ;  for  the  eye  in  this  projection  has  no  geometrical,  position 
as  it  has  in  perspective,  isometric  drawings,  etc.  This  point  is  merely  a  point  of  con- 
vergence for  all  projecting  lines.  The  eye  simply  views  the  drawing  from  above. 


145  THE   CHART.  143 

In  this  projection  it  is  possible  to  project  more  than  a  hemisphere, 
an  example  of  which  may  be  seen  in  the  Transit  of  Venus  Charts  in 
the  Nautical  Almanac  for  1882. 

This  projection  is  better  by  far  than  either  of  the  two  others.  It 
can  generally  be  made  in  a  graphic  manner  by  the  principles  of 
descriptive  geometry ;  but  when  the  author  took  up  the  subject  of 
eclipses  for  the  Nautical  Almanac,  he  found  this  method  to  be  im- 
practicable for  drawings  of  the  size  required.  He,  therefore,  devised 
a  series  of  formulae  and  tables  by  which  the  projections  can  readily 
be  made.  The  method  was  subsequently  published  in  a  pamphlet 
entitled,  Treatise  on  the  Projection  of  the  Sphere,  which  includes  both 
Stereographic  and  Orthographic  Projections. 

146.  The  Drawing. — This  and  the  kind  of  projection  to  be  selected 
may  be  determined  by  the  purposes  for  which  the  drawing  is  intended, 
or  may  be  left  to  the  judgment  of  the  computer.  When  the  author 
took  up  the  computation  of  the  eclipses,  he  adopted  the  radius  of  the 
sphere  sixteen  inches,  and  has  made  the  drawings  on  paper  22X28 
inches.  For  large  eclipses  the  scale  must  be  reduced  according  to 
the  size  of  the  eclipse ;  and  in  the  reduced  scale  the  primitive  circle 
usually  shows  in  the  drawing,  as  seen  in  the  chart,  Fig.  17,  which  is 
reduced  five-sixths — nearly  the  greatest  reduction  that  is  necessary. 
In  Fig.  17,  Plate  VI.,  the  various  geographical  positions  computed 
in  the  examples  on  the  foregoing  pages  are  marked  by  small 
circles. 

For  a  study  eclipse  or  example  I  recommend  a  chart  of  large 
scale  rather  than  small,  at  least  as  large  as  one-half  of  that  adopted 
above. 

For  plotting  the  positions  a  continually  varying  scale  is  necessary, 
and  the  most  convenient  is  a  piece  of  thin  paper  or  tracing  cloth, 
about  six  inches  long,  and  three  at  one  end  and  half  an  inch  at  the 
other,  ruled  with  converging  lines  into  ten  equal  spaces  for  degrees. 
This  can  be  folded  to  fit  between  the  ten-degree  meridians  and 
parallels,  giving  degrees,  the  minutes  to  be  estimated. 

The  charts  are  plotted  upon  the  nearest  ten-degree  meridian  to 
that  of  the  Central  Eclipse  at  Noon,  and  upon  the  parallel  of  latitude 
which  comes  nearest  to  the  middle  of  the  drawing,  giving  preference 
to  a  parallel  which  comes  nearest  to  the  central  line  in  the  above 
assumed  meridian ;  so  as  to  bring  the  central  line  near  the  assumed 
parallel  of  latitude. 


144 


THEORY   OF   ECLIPSES. 


147 


FIG.  19. 


147.  One  peculiarity  of  the  outline  curves  is  sometimes  noticed 
in  plotting  them ;  at  one  end  they  appear  to  change  their  direction 

and  curve  outward.  The  only  ex- 
planation I  can  give,  but  which  does 
not  seem  to  suit  all  cases,  is  that  on 
the  earth's  surface  they  really  do 
reverse  their  curvature.  It  is  more 

^^"^  ^^*s,  >k         \      \\  \  V 

/  \  \   \v\ vN,  apparent  when  the  pole  is  elevated, 

and  is  very  marked  in  the  eclipse 
of  Sept.  28,  1894.  In  Fig.  19, 
looking  at  the  earth  toward  the 
north  pole,  which  is  elevated,  A,  B 

is  the  fundamental  plane,  and  the  vertical  lines  represent  elements 
of  several  cones  of  penumbral  shadow.  It  is  seen  that  the  lines  on 
the  earth  approach  the  pole  (their  latitudes  increasing),  until  they 
become  tangent  to  some  circle  of  latitude,  when  they  change  their 
curvature  and  recede  from  the  pole. 

148.  For  important  total  eclipses  where  the  central  path  passes 
over  the  United  States,  special  charts  on  a  large  size  have  been  pre- 
pared and  published  in  a  joint  report  by  the  Naval  Observatory  and 
Nautical  Almanac  office.     The  two  last  of  such  publications  were  for 
the  total  eclipses  of  July  29,  1878,  and  May  28,  1900. 


SECTION   XVIII. 


PKEDICTION  FOR  A  GIVEN  PLACE. 

149.  Preliminary  Reductions. — This  section  is  the  work  of  the 
astronomer  who  has  located  himself  on  or  near  the  centre  line  to 
observe  the  eclipse.  The  latitude  and  longitude  of  the  station  must 
be  accurately  determined,  and  the  latitude  reduced  by  either  of  the 
following  formulae : 


sin  </>  =  e  sin  y 
p  sin  <pf  =  (1  —  e2)  sin  ^  sec  0 


From  the  first  of  these,  cos  </>  =  1/1  —  e*  sin2  </> 


(254) 
(255) 


149  PREDICTION   FOR   A   GIVEN   PLACE.  145 

By  reducing  and  introducing  the  following  auxiliaries : 


,,T    , 

We  have  (257) 


Equation  (254)  gives  the  reduction  to  the  geocentric  latitude,  but 
as  the  two  factors  in  equation  (257)  are  not  needed  separately,  the 
latter  is  the  most  convenient.  F  and  G  are  tabulated  in  Table  XII., 
which  is  taken  from  the  Nautical  Almanac. 

Next,  log  fa'  must  be  reduced  to  parts  of  radius. 

We  have  A  fa,  the  change  of  fa  in  1  hour  in  seconds  of  arc, 

->  the  change  of  fa  in  1  minute  in  minutes  of  arc  for  N.  A., 


3600 


and  for  the  prediction  in  parts  of  radius  we  must  have  the  last  quan- 
tity in  parts  of  radius  ;  that  is, 


=  [Lr  of  the  BESSELIAN  Tables  x  sin  1'  (258) 

When  using  addition  and  subtraction  logarithms,  the  angle  d  will 
be  needed  in  equation  (261),  which  can  be  gotten  from  the  sine  ;  but 
when  using  natural  numbers  in  equation  (274),  cos  d  is  needed,  which 
is  given  among  the  BESSELIAN  elements. 

fa'  can  be  reduced  from  the  BESSELIAN  Tables  in  the  Nautical 
Almanac  by  the  second  form  of  equation  (258),  but  it  will  be  given 
only  to  four  decimals.  By  the  first  form  it  can  easily  be  gotten  to 
five  decimals  from  the  change  of  fa  given  in  above-mentioned  table. 

In  the  examples  of  eclipses  given  at  the  end  of  the  Nautical 
Almanac,  and  which  have  been  prepared  by  various  persons,  this 
quantity,  through  some  oversight,  has  been  regarded  as  a  "  constant 
log  7.63992."  The  error  commenced  in  1886  and  has  been  con- 
tinued to  the  present  time.  It  changes  with  the  season,  and  the 
variation  is  small. 

The  quantities  log  xf,  log  yf,  as  well  also  as  log  //,  are  given  in  the 
Nautical  Almanac  to  only  four  decimals.  It  would  be  better  if  they 
were  given  to  five  for  use  with  addition  and  subtraction  logarithms. 


10 


146  THEORY  OF  ECLIPSES.  149 

In  the  examples  of  this  computation  at  the  end  of  the  Nautical 
Almanac,  natural  numbers  are  used;  and  these  logarithms  will  give 
the  numbers  sufficiently  close,  so  that  the  succeeding  terms  can  be  given 
correctly  to  five  decimals  of  logarithms.  Log  x'  and  log  y'  are  simply 
the  variations  of  x  and  y  for  one  minute,  and  they  can  be  gotten  at 
once  to  five  places  of  logarithms  from  the  differences  of  x  and  y  for 
ten  minutes  by  taking  the  logarithms  of  one-tenth  of  these  differ- 
ences. 

Five-place  logarithms  in  this  computation  are  not  as  important  as 
might  be  supposed,  for  the  resulting  quantities  are  generally  small, 
so  that  four  figures  only  are  generally  needed  in  the  final  results, 
which  four-place  logarithms  will  give  with  care. 

Each  of  the  three  assumed  times  becomes  T0  for  the  column  over 
which  it  stands. 

TQ  in  this  computation  can  be  assumed  at  pleasure,  because  x'  and 
2/'  are  the  absolute  variations  of  x  and  y,  as  explained  in  Art.  30. 

In  the  example  in  the  Nautical  Almanac,  the  longitude  co  is  called 
A  to  conform  to  the  notation  used  in  the  Use  of  the  Tables  at  the  end 
of  the  book. 

150.   General  Formula  (CHAUVENET,  Arts.  322-24). — 

#  =  to  —  u>  (259) 

A  sin  B  =  p  sin  ?'          |  ^60) 

A  cos  B  =  p  cos  <pr  cos  #  ) 

£  =  p  cos  <pr  sin  $    "} 

V  =  A  sin  (B  -  d}  V  (261) 

C  =  A  cos  (B  —  d}  ) 


m  sin 
m  cos 


mM=x  —  $)  (262) 

;os  M  =  y  —  T)  ) 

log  fj.'  =  log  [>'  of  the  BESSELIAN  Elements  X  sin  1']      (263) 

£'  =  fif  A  cos  B      ) 

T)'  =*  fi'S  sin  d  (264) 

C'       is  not  needed.  ) 


'  —  £'  j  (265) 

f  —  r  ) 


nsmN=x'  — 
n  cosN=  yf 

i  =  tan  /    [Log  tan,  Angles  of  Cones.]  (266) 


sn 


(268) 


150  PREDICTION  FOR  A  GIVEN   PLACE.  147 


T  =  dt  —      1  _  '"  ""  ^"—  (269) 

71  n 

T  =  TO  +  T     [Greenwich  Mean  Time.]  (270) 

For  an  Annular  Eclipse  and  for  Penumbra,  use  the  upper  sign.  )      /07i\ 

For  Umbral  Cone  of  Total  Eclipse,  use  the  lower  sign.  j 

Local  Mean  Astronomical  Time  T^  =  T  —  «>  (272) 

Local  Civil  Time 

If  Tv  is  less  than  12A  write  P.  M.  after  it,  and  retain  the  date. 

If  TI  is  greater  than  12*  subtract  12*  from  it,  mark  the  result 

A.  M.  and  add  1  to  the  days. 
This  latter  must  not  be  confounded  with  Standard  Time. 

If  natural  numbers  are  to  be  used,  instead  of  equations  (260)  and 
(261),  substitute  the  following  : 

£  =  p  cos  <f>f  sin  &  "| 

TI  =  p  sin  <pf  cos  d  —  p  cos  <p'  sin  d  cos  #  >  (274) 

£  =  p  sin  <pf  sin  d  +  p  cos  <pf  cos  d  cos  #  ) 

151.  Example. — First  mark  the  position  of  the  given  plan  on  the 
chart  in  the  Nautical  Almanac,  and  ascertain  by  the  dotted  lines  the 
times  of  beginning  and  ending  as  near  as  possible.  The  middle  time 
can  be  ascertained  by  comparing  the  longitude  of  the  central  line  and 
limits  as  given  to  every  five  minutes  on  another  page.  Each  of  these 
will  be  TQ  for  the  computation  following.  For  these  times  take  out 
and  interpolate  the  several  quantities  given  in  the  tables  of  BES- 
SELIAN  Elements. 

Compute  with   four-place  logarithms   carefully  to  seconds  or  a 

EXAMPLE,  PREDICTION  FOR  A  GIVEN  PLACE. 

TOTAL  ECLIPSE,  1904,  SEPTEMBER  9. 

Preliminary  Reductions. 

Assumed  position  of  the  Given  Place  ^  =  —  11°  54'  0"    w  ==  120°  0'  0"  W. 

Reduced  latitude  log  p  sin  f  =  —  9.31141        log  p  cos  <$>'  +  9.99062. 

From  the  Eclipse  Chart  and  Longitude  of  the  centre  line. 

Beginning  8*  2".  Middle  9*  31™.  Ending  10*  52M. 

For  these  times  from  the  BESSELIAN  Elements  in  the  Nautical 
Almanac. 

x  —0.44223                   +0.38516                   +1.13807 

y  —0.03315                    —0.28973                    —0.52331 

//,  +0.53258                    —0.01373                    +0.53255 

^,  121°  11.40                    143°  26.85                   163°  42.25 
A//  from  the  table  15°  0'  18"  =  54018",  whence  log  //'  =.-  7.63996 


148 


THEORY   OF   ECLIPSES. 


151 


The  other  quantities  in  the  logarithms  vary  slowly ;  they  are  given 
in  the  example  where  they  are  needed. 


Computation  for  the  Times. 


TO 

gA   2m 

Qh    31m 

10*  52"» 

(259)  N.  A.  //! 

121  11  24 

143  26  51 

163  42  15 

u 

120  0  0 

#  =  ^—0) 

4-1  11  24 

4-  23  26  51 

+  43  42  15 

sin  •& 

4-  8.31739 

9.59979 

+  9.83943 

COS# 

4-  9.99991 

9.96257 

+  9.85909 

(260)  A  sin  B  =  p  sin  $' 

—  9.31141 

p  cos  $' 

+  9.99062 

A  cos  B  —  p  cos  0X  cos  # 

4-  9.99053 

4-  9.95319 

+  9.84971 

tan  B 

9.32088 

9.35822 

9.46170 

B 

—  11  49  28 

—  12  51  7 

-16  8  51 

cosB 

9.99069 

9.98898 

9.98252 

log  A 

+  9.99984 

9.96421 

+  9.86719 

(261)  N.  A.  from  sin  d  d 

4-  5  15  42 

5  14  20 

-t-  5  13  6 

B  —  d 

-17  5  10 

—  18  5  27 

—  21  21  57 

sin  (B  —  d) 

—  9.46807 

—  9.49210 

—  9.56148 

cos  (.B  —  d) 

+  9.98040 

4-  9.97798 

+  9.96907 

(261)  ^=  p  cos  $'  sin  # 

4-  8.30801 

4-  9.59041 

+  9.83005 

'  N.  A.  « 

—  9.64565 

4-  9.58564 

+  0.05617 

I  —  I 

A  1.33764 

B  0.00477 

B  0.22612 

AB 

1.35716 

8.04300 

9.83451 

m  sin  M=x  —  £ 

—  9.66517 

—  7.62864 

+  9.66456 

(261)  N.  A.  y 

—  8.52048 

—  9.46199 

—  9.71876 

77  =  A  sin  (B  —  d) 

—  9.46791 

-9.45631 

—  9.42867 

1  —  I  B 

0.94743 

0.00568 

0.29009 

A 

0.89542 

8.11950 

9.97784 

m  cos  M=  y  —  y 

4-  9.41590 

—  7.57581 

—  9.40651 

(262)       tan  M 

0.24927 

0.05283 

0.25805 

M 

—  60  36  29 

—  131  31  25 

+  118  53  58 

sin  M 

9.94016 

9.87429 

9.94224 

log  m 

4-  9.72501 

7.75435 

+  9.72232 

(263)  N.  A.  log  //! 

+  7.63996 

(264)  N.  A.  a/ 

4-  7.9683 

4-  7.9683 

+  7.9682 

%'  =  p'  A  cos  J5 

4-  7.6305 

4-  7.5931 

+  7.4897 

Z  —  Z.B 

0.3378 

0.3752 

0.4785 

^ 

0.0707 

0.1375 

0.3031 

n  sin  N=x/  —  £' 

4-  7.7012 

4-  7.7306 

+  7.7928 

(264)  N.  A.  y' 

—  7.4597 

—  7.4599 

—  7.4600 

N.  A.  sin  d 

4-  8.96238 

4-  8.96050 

+  8.95880 

J?/  =  /u'  £  sin  d 

4-  4.91035 

4-  6.19087 

+  6.42881 

Z  —  Z^ 

2.5494 

1.2690 

1.0312 

J? 

0.00123 

1.29177 

1.06984 

ncos  N=y'  —  n' 

—  7.4609 

—  7.4826 

—  7.4986 

151 


PREDICTION   FOR  A   GIVEN  PLACE. 


149 


(265) 

tanN 

0.2403 

0.2480 

0.2942 

N 

+  119  54    2 

+  119  27  50 

+  116  55  38 

sin  N 

9.9380 

9.9398 

9.9502 

logw 

+  7.7632 

7.7908 

+  7.8426 

(267) 

N.  A.     I  /! 

+  9.72638 

—  8.13767 

+  9.72636 

(266) 

N.  A.     t  =  tan/ 

-f  7.66686 

+  7.66470 

+  7.66687 

(261) 

C  =  4cos  (B  —  d) 

9.98024 

9.94219 

9.83626 

it 

+  7.64710 

-f  7.60689 

+  7.50313 

I  —  I 

B  2.07928 

A  0.53078 

B  2.22323 

AB 

0.00363 

0.64291 

0.00260 

logL 

+  9.72275 

—  8.24980 

+  9.72376 

(268) 

M—N 

—  180  30  31 

—  250  59  15 

+  1     58  20 

log  m 

+  9.72501 

+  7.75435 

+  9.72232 

log  (1  :  ») 

+  2.2368 

+  2.2092 

+  2.1574 

sin  (M—N) 

+  7.94826 

+  9.97564 

+  8.53675 

cos  (M—N) 

—  9.99998 

—  9.51292 

+  9.99974 

m  sin  (     ) 

-f  7.67327 

+  7.72999 

+  8.25907 

i 

,:n  $  _m  sin  (M—N) 

-f  7.95052 

+  9.48019 

+  8.53531 

sin  y 
Lt 

(271) 

1> 

f  +  179  29  19 

+  162  24  53 

+  1     57  57 

\            / 

\ 

+    17  35    7 

(269) 

cos  V 

—  9.99998 

=F  9.97922 

+  9.99974 

log  (1) 

—  1.9595 

±  0.4382 

+  1.8809 

(2) 

—  1.9618 

—  9.4765 

+  1.8795 

(271) 

Nos.  (1) 

+  (!)  —  91.10 

(l)q=2.74      + 

(1)+  76.01 

-(2) 

+  91.58 

+  0.30 

—  75.77 

r 

{  +   0.48 

2.44 

+    0.24 

I 

+  3.04 

(270) 

Greenwich  Mean  Time 

T         f  8*  2"  48 

9ft  28^.56 
9  34    04 

10*  52".  24 

(272)  Longitude  in  time  «  8   0.00  W. 

Local  Astronomical  Time  2\      0   2.48  1  28.56  2  52.24 

34.04 

(273)  Local  Civil  Time  Sept.  9,          12*  2TO.48  P.M.    1*  28W.56  P.M.   2*  52™.24  P.M. 

34  04 

"     Middle  1   31   30P.M. 

Duration  of  Totality,  double  of  term  (1)  5m.48  — 5*  288.8. 

fraction  of  a  minute,  or  preferably  to  five-place  logarithms ;  but  as 
some  of  the  quantities  in  the  Almanac  are  given  only  to  four  places 
of  logarithms,  a  portion  of  the  work  must  be  computed  with  this 
number.  With  addition  and  subtraction  logarithms  small  terms  in 
four-place  logarithms  may  be  used,  and  the  work  continued  with 
five  places. 

When  computing  the  total  phase,  the  quantities  x'  —  £'  and  y'  —  rf 
must  necessarily  be  very  small,  since  the  observing  station  is  se- 
lected near  the  central  line.  Four-place  logarithms  will  be  suffi- 


150  THEORY  OF   ECLIPSES.  151 

cient;  but  the  angle  ^will  be  a  little  uncertain,  but  in  the  end  this 
does  not  matter,  for  r  will  be  very  small.  The  three  columns  being 
computed  simultaneously,  I  and  log  i  for  penumbra  must  be  used  in 
the  first  and  third  columns,  and  ^  and  log  \  for  the  umbra,  in  the 
middle  column,  which  latter  will  give  the  times  of  totality. 

The  angle  <p  has  two  values  as  in  all  the  foregoing  problems. 
Here,  however,  we  must  take  it  obtuse,  with  its  cosine  negative  in 
the  first  column  for  beginning  of  penumbra,  and  acute,  with  its  cosine 
positive  for  ending  in  the  third  column.  In  the  second  column  the 
double  sign  merely  counteracts  the  negative  sign  of  L  and  take  both 
values,  which  give  the  beginning  and  ending  of  totality.  Quantities 
for  the  umbra  generally  lie  so  near  together  that  in  the  first  approxi- 
mation, which  may  also  be  the  final,  one  computation  answers  for  both. 

CHAUVENET  omits  to  state  that  the  negative  sign  of  L  for  the 
umbra  of  total  eclipse  necessitates  a  change  of  sign  of  equation  (268), 
and  of  the  first  term  of  equation  (269),  which  omission  is  here  cor- 
rected by  the  condition  (271).  And  in  the  example  this  is  noted  as 
—  (1),  referring  to  the  first  term  of  the  formula  for  the  total  shadow. 
The  two  values  of  (p  after  changing  the  sign  give  rise  to  the  double 
sign,  the  upper  for  beginning  and  the  lower  for  ending,  as  stated 
above.  By  this  condition  T  for  beginning  of  totality  properly 
becomes  negative,  and  for  ending,  positive.  In  the  example  f  is 
not  computed  immediately  after  ^,  because  it  is  not  needed  until 
equation  (267)  is  reached,  when  it  is  taken  up.  This  is  one  of  the 
few  cases  in  which  it  is  preferable  to  take  up  any  portion  of  the 
formulae  out  of  the  order  given  in  this  work.  The  example  given  in 
full  will  doubtless  explain  the  formula  more  fully. 

152.  Second  Approximation.  —  This  may  not  always  be  necessary. 
If  the  times  have  not  been  well  chosen  to  start  with,  T  may  result  as 
a  quantity  of  several  minutes,  in  which  case  a  repetition  may  be 
unavoidable,  taking  as  TQ  the  resulting  times  or  some  integral  min- 
ute near  them  ;  and  the  results  then  will  be  correct  within  the  frac- 
tion of  a  second.  In  this  example  it  seems  that  no  repetition  is 
necessary.  The  times  were  very  carefully  chosen,  and  the  methods 
employed  will  be  noticed  in  Art.  157  on  checks  to  this  com- 
putation. 

The  following  formula  is  given  by  CHAUVENET  as  a  substitute  for 
Nos.  (268),  (269)  : 

T=  ~  -  sin(^"~  N~  ^)  (275) 


n  sn 


152 


PREDICTION   FOR  A   GIVEN   PLACE. 


151 


which,  he  remarks,  in  the  second  approximation  will  be  more  conve- 
nient, but  when  (p  is  small  will  not  be  so  precise. 

153.  Angles  of  Position. — These  are  required  for  the  penumbra  to 
guide  the  observer  to  that  part  of  the  sun's  disk  where  the  first  con- 
tact will  appear,  so  that  he  may  concentrate  his  attention  upon  this 

point  to  note  the  earliest  moment  of  the  first  contact. 
< 
Q  =  Angle  of  position  from  the  North  point  toward  the  East 

(Art.  80). 
V  =  Angle  of  position  from  the  Vertex  toward  the  East. 

Q  =  N+4  (276) 

>«nr-*  +  »*n  (277) 

p  cos  Y  =  "n  +  *T  3 

V=Q-r  (278) 

=sJV+^.-r  (279) 

Or  else  independently  of  the  eclipse  formulae, 


(280) 


sin  C  sin  q  =  p  cos  <f>r  sin  #  ) 

sin  C  cos  q  =  p  sin  <p?  cos  d  —  p  cos  <?'  sin  d  cos  #  ) 
in  which  C  =  Zenith  distance. 

q  =  Parallactic  angle  =  y  of  equation  (277). 

These  last  formulas  are  essentially  the  same  as  equation  (277). 

From  the  foregoing  computation  for  the  final  times  we  have  the 
angles  of  position  from  the  north  point. 

(276)  Q  =  N+<p.     For  beginning,  +  299°  23'  =  60a  37'  to  W. 
For  ending,        -f  118°  54'. 


EXAMPLE,  ANGLES  FKOM  THE  VERTEX. 


(277) r 


Beginning. 

Ending. 

Beginning. 

Ending. 

r 

4-  9.6812 

4-  9.3802 

(277)  n 

—  9.4679 

—  9.4287 

nf 

4-  4.9103 

4-  6.4288 

£ 

-f-  8.3080 

4-  9.8301 

r)'  r 

4-  4.5915 

+  5.8090 

?' 

+  7.6305 

4-  7.4897 

I  —  IA 

4.8764 

3.6197 

S'T 

+  7.3117 

6.8699 

B 

0.0000 

0.0001 

I  —  IB 

0.9963 

2.9602 

pcosy 

—  9.4679 

—  9.4288 

A 

0.9501 

0.0005 

psiny 

4-  8.2618 

4-  9.8296 

tan  y 

8.7939 

0.4008 

7 

4-  176°  26' 

+  111°40/ 

(278)  Angle  from  the  vertex  V=Q  — 


122    57 


7    14 


The  angle  from  the  north  point  is  used  in  telescopes  equatorially 
mounted,  but  for  others  the  angle  from  the  vertex  is  more  convenient, 
which  is  the  point  of  the  sun  nearest  to  the  zenith  of  the  observer. 


152  THEORY   OF   ECLIPSES.  154 

154.  Partial  Eclipse;  Greatest  Eclipse.  —  This  article  is  for  an 
eclipse  partial  at  the  station  of  the  observer,  though  it  may  or  may 
not  be  total  or  annular  elsewhere.  The  times  are  computed  for  the 
penumbra  exactly  as  in  the  foregoing  example,  but  in  order  to  get 
the  middle  of  the  eclipse  and  the  magnitude  correctly,  it  will  be 
necessary  to  make  the  first  computation  for  a  time  as  near  the  middle 
as  possible  ;  or  if  the  times  can  be  known  as  closely  as  in  the  above 
example,  the  three  columns  can  be  carried  on  together.  A  second 
approximation,  however,  is  a  great  guard  against  mistakes.  The 
time  of  the  middle  of  the  eclipse  and  of  greatest  eclipse  are  synony- 
mous. The  formulae  are  derived  in  the  same  manner  as  those  in 
Art.  78. 


T,  =  T0  +  r  (282) 

The  quantities  for  these  equations  must  be  taken  from  the  first 
approximation,  or  for  a  time  near  the  middle  ;  within  ten  minutes 
may  be  near  enough.  The  term  for  r,  as  noted  in  Art.  78,  is  already 
given  in  the  computation.  In  the  above  example  we  have 

TI  =  -f  Om.30          G.  M.  T.     Tt  =  9*  31m.30 

The  above  equations,  if  applied  to  the  ends  of  an  eclipse,  may  not 
give  the  time  of  the  middle  within  fifteen  minutes,  nor  will  the  mean 
of  these  be  anywhere  near  the  correct  time. 

The  middle  time  as  computed  for  prediction  is  not  equidistant 
from  the  times  of  beginning  and  ending.  This  is  caused  by  the 
rotation  of  the  earth,  for  when  the  end  of  the  eclipse  is  in  the  after- 
noon, as  in  the  present  example,  the  surface  of  the  earth  at  the 
observing  station  will  be  much  more  inclined  to  the  fundamental 
plane  than  at  the  beginning  ;  the  shadow  will  move  more  rapidly, 
and  the  end  occur  sooner  than  otherwise.  If  an  eclipse  is  computed 
for  the  local  morning  time,  the  beginning  will  be  nearer  to  the 
middle  time  than  the  ending. 

155.  Duration.  —  For  total  or  annular  eclipse  this  is  given  by 
doubling  the  first  term  of  equation  (269)  for  r  taken  as  positive 

(283) 


It  is  also  simply  the  difference  between  the  beginning  and  ending 
of  totality. 


156  PREDICTION   FOR  A   GIVEN   PLACE.  153 

156.  Magnitude.  —  The  formulae  are 

A  ==  ±  m  sin  (M  '—  N)  =  =b  L  sin  ^  (284) 


&  =  0.2723  (286) 

Or,  approximately,  omitting  the  augmentation  of  the  moon's  semi- 
diameter, 

e=^b)  •      (287> 

Jtf=  ez]=  e  sin^  (288) 

A  is  always  to  be  taken  as  positive,  and  M  results  as  a  fraction  of 
the  sun's  diameter.  In  the  last  equation  the  numerical  difference  is 
to  be  used  for  the  double  sign.  In  all  these  equations  the  quantities 
should  be  taken  for  some  time  near  the  middle  to  give  correct  results. 
After  computing  A  by  logarithms,  or  taking  it  from  the  computation, 
the  rest  of  the  work  is  more  easily  performed  with  natural  numbers, 
the  division  requiring  only  three  decimals.  L  has  to  be  computed  or 
taken  from  the  computation.  And  for  a  partial  eclipse  L\  for  the 
umbral  cone  has  not  been  used,  so  that  the  second  form  of  equation 
(285)  must  be  substituted.  Jc  is  the  constant  used  in  computing 
the  eclipse  tables  (Art.  20),  and  /  for  equation  (287)  is  given  in  the 
BESSELIAN  Tables.  In  formula  (288)  the  numerical  difference  gives 
M.  These  formulae  are  simple,  and  no  example  is  needed  for  the 
present  eclipse. 

157.  Checks  on  the  Times.  —  It  is  stated  at  the  end  of  the  Nautical 
Almanac  in  the  examples  of  eclipses,  that  the  positive  cosine  of  <p  for 
beginning  and  the  negative  sign  for  ending  will  furnish  "inaccurate" 
times  for  beginning  and  ending.  It  is  true,  but  quite  as  true  also 
that  they  are  utterly  worthless  for  any  purposes  of  check  or  compari- 
son, for  they  may  differ  fifteen  minutes  from  the  correct  times. 

A  rigorous  check  upon  the  times  may  be  had  from  the  simple 
statement  repeated  several  times  in  this  work, 

#  =  the  Local  Apparent  Hour  Angle.  (289) 

We  have  the  hour  angle  for  the  assumed  times,  $.  Correct  this  for 
the  true  times  and  apply  the  equation  of  time,  and  we  have  the  equality, 

T=  r90  +  Equation  of  Time  +  r.  (290) 


154 


THEORY   OF   ECLIPSES. 


157 


Here  the  equation  of  time  is  to  be  taken  from  the  Nautical  Alma- 
nac, page  L,  for  the  month,  and  interpolated  for  the  sun's  apparent 
hour  angle,  for  which  the  Greenwich  mean  time  will  suffice  within  a 
fraction  of  a  second.  For  the  correction  of  &  for  the  computed  times 
this  depends  upon  the  variations  of  fjtl}  which  varies  15°  per  hour 
of  time,  or  nearly  that  within  small  limits,  so  that  the  variation  of  //x 
will  be  equal  to  the  change  in  the  time,  which  is  T.  The  equation  of 
time  is  given  in  seconds,  so  applying  this  first  and  reducing  to  minutes 
before  applying  r,  we  have  for  the  present  example  the  following : 

CHECK  ON  THE  TIMES  BY  #,  FORMULA  (290). 

Beginning. 

Hour  angles  for  the  assumed  times      11°  IV  24 
"  in  time  0A  4™  45*.6 

Equation  of  Time  (Nautical  Almanac)  —  2    45  .91 

Sum  1°    1    59'7 

\0   1.99 

T!  computed  -f  0   0.49 

Corrected  hour  angle  =  &  0    2.48 

Computed  local  times  12   2.48 

Fig.  20  (reduced  to  half  size)  is  a  plot  of  part  of  the  path  of  this 
eclipse  made  on  computing  paper,  to  take  advantage  of  the  ruled 

FIG.  20. 


Middle. 
23°  26'  51 

Ending. 
43°  42'  15" 

!»  33™  41^4 

2*  54m  498.0 

—  2    41.17 

—  2    48.37 

1    31      0.2 

2   52      0.6 

1    31.00 

2   52.01 

+  0.30 
1   31.30 

+  0.24 
2  52.25 

1   31.30 

2   52.24 

13° 


lines ;  the  degrees  of  latitude  and  longitude  form  nearly  a  square, 
and  though  not  in  proper  proportions,  yet  they  serve  the  purpose 


157  PREDICTION   FOR  A   GIVEN   PLACE.  155 

intended.  The  place  from  which  the  above  prediction  was  computed 
was  assumed  on  this  plot,  and  is  marked  P.  The  lines  across  the 
path  are  the  positions  at  9A  30m  and  9A  35m,  taken  from  the  Nautical 
Almanac  and  carefully  plotted.  After  the  computation  was  fin- 
ished, the  times  of  totality  were  checked  as  follows  :  For  the  middle 
time  with  a  scale  of  equal  parts  (50  to  an  inch  was  used)  aP  =  58, 
ab  =  227,  whence  the  ratio  is  0.255,  being  a  fraction  of  five  minutes, 
and  the  time  of  the  middle  of  the  eclipse  at  P  is  31m.27.  Omitting 
the  decimal,  we  have  the  assumed  time  of  the  middle  for  the  compu- 
tation. The  correct  time  is  31W.30,  which  gives  the  correct  fraction 
of  the  distance  aP  =  0.260,  and  interpolating  the  duration  in  the 
Nautical  Almanac  for  this  fraction  we  have  for  the  central  line  5m 
45M.  Now  if  a  semicircle  be  described  in  the  centre  line  tangent 
to  the  limiting  curve,  the  duration,  which  is  0  at  the  limits,  will,  at 
the  point  P,  be  in  proportion  to  that  in  the  centre  line  as  the  cord 
of  the  circle  is  to  the  diameter.  This  is  easily  gotten  as  follows : 
PC  measured  by  scale  (50  parts  to  the  inch)  gives  31,  cd  likewise 
111  ;  hence  the  ratio  0.2793,  which  is,  in  fact,  sin  $  of  the  former 
computation,  log  sin  =  9.4461,  from  which 

<P  16°  13' 

cos  </>  9.9824 

log       5m  45M  =  345M         2.5379 

Duration  at  P  5    31.3=331.3         2.5203 

"        as  computed  5    28.8 

Difference  as  a  check  0      2  .5 

Having  checked  the  middle  time  and  the  duration,  the  other  times 
of  totality  are  completely  checked  also.  The  results  are  remarkably 
close  for  measurements  by  scale.  The  closeness  of  the  results  of  the 
umbra  result  greatly  from  Fig.  20  being  drawn  on  a  very  large  scale, 
on  which  a  small  fraction  of  a  minute  can  be  measured,  and  partly 
from  great  care  in  plotting,  and  partly  perhaps  from  chance.  For 
the  penumbra  great  care  was  taken  to  locate  the  point  on  the  chart 
in  the  Nautical  Almanac,  and  for  beginning,  taking  a  scale  of  six 
equal  parts  whose  length  is  a  mean  of  the  distance  between  the 
7A  and  8A  and  between  the  8A  and  9*  curves.  For  the  ending,  as  the 
shadow  is  moving  faster,  the  scale  was  taken  greater  than  the  dis- 
tance between  the  10*  and  11A  curves,  which  could  only  be  estimated. 
These  means  of  verification  are  open  to  any  one  who  chooses  to  avail 
himself  of  them.  No  previous  computation  was  made  before  the 
example  given  above. 

The  lines  drawn  from  P  to  the  centres  of  the  circles  make  with 


156  THEORY   OF   ECLIPSES.  157 

the  meridian  approximately  the  angles  of  position  of  the  points  of 
contact  of  totality,  but  not  exactly,  because  these  angles  are  meas- 
ured in  the  fundamental  plane  or  a  parallel  to  it ;  whereas  the  plane 
of  Fig.  20  is  tangent  to  the  earth's  surface  and  much  inclined,  the 
lower  right  hand  corner  being  the  lowest  point. 

These  angles  are  as  follows,  from  the  example  Art.  151  ;  the 
negative  sign  in  equation  (268)  being  used  in  accordance  with  the 
condition  (271),  which  is  here  shown  to  be  correct. 

Beginning  of  Totality.  Ending. 

N        +  119  28  -f  119  28 

<!>          +  162  25  +    17  35 
Q           (  281  53  to  E.  137     3  to  E. 

1    78    7toW. 

These  angles  will  be  again  referred  to  in  Art.  165  of  the  next 
section.  The  angle  N,  w^hich  the  path  makes  with  the  principal 
meridian,  cannot  be  shown  in  this  figure  except  approximately  on 
account  of  the  inclination  of  the  plane  of  the  drawing,  and  also 
because  the  meridians  are  not  parallel  to  the  principal  meridian. 

These  angles  are  usually  not  wanted  for  the  umbral  phase,  but  are 
given  here  for  a  better  understanding  of  the  eclipse. 


SECTION    XIX. 

PKEDICTION  BY  THE  METHOD  OF  SEMIDIAMETEES. 

158.  THE  present  section  was  not  included  in  the  original  design 
of  the  present  work,  but  is  an  after-thought  suggested  by  the  fact 
that  CHAUVENET'S  treatise  gives  no  idea  how  the  eclipse  appears  to 
the  eye.  Fig.  21  of  this  section  shows  the  actual  relative  motions 
of  the  observer  and  the  shadow,  which  are  not  shown  in  the  previous 
orthographic  projections.  And  in  the  latter  part  of  the  section,  the 
computation  is  based  upon  the  supposition  of  direct  vision.  For  a 
student  who  finds  CHAUVENET'S  treatise  too  difficult,  this  section 
will  be  found  an  excellent  preparatory  study.  It  requires  no 
knowledge  of  the  preceding  portions  of  this  work. 

For  this  section  the  author  is  chiefly  indebted  to  LOOMIS'  Practi- 
cal Astronomy,  an  excellent  work,  though  this  portion  of  it  is  not 
original  with  him,  being  the  method  of  the  ancient  astronomers. 


159    PREDICTION  BY  METHOD  OF  SEMIDIAMETERS.     157 

159.  Projection.  —  The  data  for  this  are  simply  the  elements  of  the 
eclipse  given  in  the  Nautical  Almanac,  together  with  the  Equation 
of  Time  in  the  body  of  the  Almanac.  The  data  for  the  present 
example  are  therefore  found  in  Art.  21  of  this  work. 

The  radius  of  the  earth  is  taken  as  the  diiference  of  the  parallaxes 
of  the  sun  and  moon,  which  has  been  styled  the  relative  parallax. 
The  sun  and  moon  are  seen  from  the  earth  at  their  apparent  relative 
sizes,  and  their  angular  dimensions,  their  semidiameters,  may  be  taken 
as  linear  measurements  in  seconds  to  any  scale  and  laid  down  upon 
a  drawing.  The  earth  and  moon,  on  the  contrary,  are  not  seen  from 
the  sun  according  to  their  actual  proportions,  because  they  are  at 
different  distances,  and  we  must  find  their  apparent  proportions  as 
seen  from  the  sun.  If  we  regard  the  cone  of  solar  parallax,  its  radius 
x,  at  the  distance  of  the  moon  from  the  sun,  will  be  in  proportion  to 
the  earth's  actual  radius,  E,  as  the  distances  of  the  moon  and  earth  are 
from  the  sun.  That  is 

x  _r'  —  r 
E~       rf 

11  11 

And  since  r  =  --  =  -  and  r'  =  —    —  =  —  nearly 

sin  TT       TT  sin  TT'        it' 


n  —  rJ  is  then  the  radius  of  this  cone  of  parallax  at  the  distance  of 
the  moon  from  the  sun,  and  the  moon's  radius  and  this  quantity  can 
also  be  regarded  as  linear  quantities  ;  hence,  x  —  rJ  for  the  earth,  s' 
for  the  sun,  and  s  for  the  moon,  are  now  all  reduced  to  the  same 
linear  scale.  This  quantity  TT  —  TZ'  is  the  actual  radius  of  the  earth's 
sphere  used  throughout  the  whole  of  this  work,  which  can  easily  be 
shown  from  equation  218,  Art.  113,  namely  — 


This  equation  gives  y  at  the  time  of  conjunction  as  a  decimal  fraction, 
of  which  the  denominator  must  be  unity,  the  radius  of  the  earth  ; 
and  the  value  of  b  we  have  from  Art.  25,  therefore  the  denominator 
becomes 

T  -  TT')  (293) 


sn 
which  agrees  with  the  quantity  gotten  above. 


158  THEORY   OF   ECLIPSES.  160 

160.  Data  for  the  Projection. — The  quantities  following  must  be 
taken  from  the  Nautical  Almanac  and  reduced  as  here  shown.    They 
are  the  elements  of  the  eclipse,  given  also  in  Art.  21  of  the  present 
work.     The  nearest  second  of  arc  is  close  enough. 

G.  M.  T.  of  cT  in  R.  A.  8*     49™  34* 

Longitude  of  the  place  (west)  800 

Local  Astronomical  Time  0       49  34 

"     Civil  Time  12       49  34 

Equation  of  Time  (Mean  to  Apparent)  +2  47 

Local  Apparent  Time  12       52  21 

Sun's  Declination     d'  or  d  +  5°     14'  56" 

Moon's  Declination  5  +  5         4  31 

Moon  south  of  the  sun  in  this  example  0       10  25 

Difference  of  Hourly  Motions  in  R.  A.  137M6  =  34  17 

This  is  to  be  reduced  to  the  arc  of  a  great  circle  by  multiplying  it  by 
the  cosine  of  the  moon's  declination,  which  gives  2049". 

Difference  of  Hourly  Motions  in  Declination         —  636" 

Moon's  parallax,  3683",  to  be  reduced  to  the  latitude  of  the  given 
place,  — 11°  56',  by  multiplying  it  by  the  earth's  radius  for  the 
place.  This  may  be  taken  from  Table  IV.  In  the  present  exam- 
ple the  earth's  radius  is  9.9999,  which  hardly  changes  the  value. 

Hence,  the  Reduced  Parallax  3682" 
Sun's  Parallax  9 

Relative  Parallax  3673. 

Sun's  Semidiameter  953. 

Moon's  Semidiameter  1004. 

Geographical  latitude  of  given  place  is  — 11°  54',  which  must  be 
reduced  to  the  geocentric  latitude,  as  in  Art.  149,  or  the  angle  of  the- 
vertical  may  be  numerically  subtracted,  taken  from  Table  XII. 

Hence,  the  Reduced  Latitude  for  the  Place,  11°  49' 

161.  With  the  relative  parallax  3673",  and  assuming  a  scale  of 
1000"  to  an  inch,  describe  the  circle  ADBC  (Fig.  21,  Plate  VIIL), 
which  will  represent  the  earth's  sphere.    AB  and  CD  are  the  princi- 
pal axes,  as  in  the  former  projections.     The  earth  is  supposed  to  be 
seen  from  the  sun,  which  is  therefore  vertically  over  the  centre  of 
the  circle  Z   at  apparent  noon  ;  and  this  therefore  determines  the 
positions  of  the  parallels  of  latitude  on  the  earth.     It  is  also  very 
nearly  vertically  over  any  other  part  of  the  drawing. 


161     PREDICTION  BY  METHOD  OF  SEMIDIAMETERS.     159 

The  path  of  the  moon  will  now  be  drawn.  As  it  is  625"  south  of 
the  sun  at  conjunction,  set  off  this  amount,  ZK,  below  the  point  Z, 
on  the  axis  of  Y,  which  is  the  meridian  of  conjunction  in  right 
ascension  at  local  apparent  noon.  Lay  off  ZH  along  the  axis  of 
X  equal  to  2049",  the  difference  of  the  hourly  motions  of  the  sun 
and  moon  in  right  ascension  reduced  to  the  arc  of  a  great  circle,  and 
HI  perpendicular  to  it  and  equal  to  —  636",  the  difference  of  the 
hourly  motions  in  declination.  ZI  will  then  represent  the  direction 
of  the  moon's  path  and  the  distance  of  its  motion  in  one  hour. 
Through  K  draw  an  indefinite  line  parallel  to  ZI9  which  will  be 
the  actual  path  of  the  moon's  centre. 

The  hour  points  on  the  path  must  be  so  laid  off  that  the  time  of 
conjunction  at  apparent  noon  ;  that  is,  52m  21*  —  52m.35  after  the  12- 
hour  point,  shall  fall  on  the  point  K.  ZI  measures  by  scale  2146", 
which  is  the  motion  in  one  hour,  or  60',  so  we  make  the  proportion 

60m.OO  :  52^.35  :  :  2146  :  1872 

Laying  off  the  distance,  1872,  of  the  scale  toward  the  left  of  K  gives 
the  12-hour  point  of  the  path.  The  other  hours  are  found  by  laying 
off  on  each  side  of  this  the  distance,  ZI.  The  path  can  now  be 
divided  into  10-minute  spaces,  and  into  minutes  where  necessary. 

162.  The  parallel  of  latitude  of  the  given  place  is  now  to  be  pro- 
jected, which  will  be  an  ellipse.  The  axes  may  be  found  in  two  ways, 
the  first  of  which  is  wholly  geometrical.  Revolve  the  north  pole 
around  the  line  CD  as  an  axis  down  into  the  fundamental  plane;  the 
south  pole  will  fall  on  the  left  of  the  axis  at  P'.  The  semicircle 
CA  D  will  fall  into  the  right  line  CZD.  The  right  line  CZD  will 
become  the  semicircle  C B  Dy  the  sun  being  then  in  the  line  ZBy 
produced ;  and  its  declination  being  north,  the  equator  will  become  a 
right  line,  E'  E",  with  the  arc  B  Ef,  equal  to  the  sun's  declination, 
the  equator,  E'  E",  will  be  perpendicular  to  the  axis  Z  P',  and  the 
given  parallel  will  be  the  line  a  b,  11°  49'  south  of  the  equator. 
Now  revolve  the  axis  of  the  earth  to  its  normal  position.  Pf  will 
fall  below  the  fundamental  plane  out  of  sight.  Er  will  fall  at  E9  and 
the  ellipse  A  E  B  may  be  drawn  if  desired  ;  the  point  b  of  the  given 
parallel  will  fall  at  d  on  the  upper  surface  of  the  sphere,  while  a 
will  fall  at  c  on  the  under  surface ;  c  d  is  the  conjugate  axis  of  the 
ellipse ;  bisecting  this  e  is  the  middle  point,  which  is  also  the 
revolved  position  in  which  this  parallel  intersected  the  axis  of  the 
earth.  Through  e  the  transverse  axis  mn  can  be  drawn  at  right 


160  THEORY   OF   ECLIPSES.  162 

angles  to  c  d;  the  length  of  the  semiaxis  is  given  by  the  half  length 
of  the  diameter  a  b,  which  is  shown  in  its  true  length.  Upon  these 
axes  the  ellipse  can  be  described. 

The  second  method  is  more  accurate,  as  the  axes  are  partly  com- 
puted. It  is  seen  from  the  geometry  of  the  figure  that  Z d  is  in 
projection  the  angle  <pr  —  d  and  Zc  the  angle  tpf  +  d,  and  that  the 
half  length  of  a  b  is  sin  <p,  each  in  proportion  to  the  radius  Z  B. 
Hence,  we  have 


(294) 


Distance  of  the  two  ends  of  the  conjugate  axis  from  the  centre 

of  the  sphere,  p  sin  (<?'  ±  d)  * 

Length  of  sera itrans verse  axis,    p  cos  y 
(+  for  the  northern  point,  —  for  the  southern,  in  all  cases.) 

d  is  the  sun's  declination,  or,  more  properly,  the  quantity  used  in  the 
eclipse  formula?,  whose  sine  and  cosine  are  given  in  the  BESSELIAN 
tables  of  the  Nautical  Almanac. 
In  the  present  case  we  have 

Northerly  point          Zc  =  3673  sin  (—    6°  44')  =  —    431" 
Southerly  point          Zd  =  3673  sin  (—16     54  )  =  —  1068 
Semitransverse  axis  mn  —  3673  cos  ( — 11     49)=       3595 

From  these  figures  the  ellipse  can  be  constructed.  The  best 
method  to  be  used  in  this  case  is  by  drawing  two  circles,  one  upon 
each  axis.  From  any  point  (/)  of  the  outer  circle  draw  a  line  to 
the  centre,  cutting  the  smaller  circle  at  the  point  g.  From  /  and  g 
draw  lines  parallel  to  the  axes,  and  where  they  intersect  at  h  will  be 
a  point  of  the  ellipse ;  any  number  of  points  can  be  gotten  in  the 
same  manner,  and  the  advantage  of  this  construction  is  that  if  the 
outer  circle  is  divided  into  six  equal  portions  of  15°  each,  these 
points  will  give  the  hours  1,  2,  3,  etc.,  before  and  after  the  noon 
point  on  the  axis.  Intermediate  points  can  be  gotten  if  desired,  for 
the  divisions  for  equal  times  will  be  unequal  on  the  ellipse. 

This  projection,  depending  upon  the  true  sun,  gives  local  apparent 
time ;  d  is,  therefore,  the  position  of  the  sun  at  apparent  noon,  and 
the  integral  hours  can  be  laid  oif  by  the  method  described  above.  6A 
A.  M.  will  be  the  point  m  and  6A  P.  M.  the  point  n;  sunrise  and 
sunset  will  be  the  points  where  the  ellipse  is  tangent  to  the  earth's 
circle ;  in  the  present  case  very  near  m  and  n.  The  ellipse  can  be 
divided  into  10-minute  spaces  for  that  portion  along  which  the  eclipse 
occurs. 

*  Treatise  on  the  Projection  of  the  Sphere,  Including  Orthographic  and  Stenographic 
Projections,  by  the  author  of  the  present  work,  p.  12. 


163   PREDICTION   BY   METHOD   OF   SEMIDIAMETERS.   161 

163.  The  times  can  now  be  found  as  follows  :  Take  the  sum  of 
the  semidiameters  1957",  in  the  dividers,  and  running  the  left  leg  on 
the  moon's  path,  and  the  right  leg  on  the  ellipse,  find  the  points 
where  they  both  mark  the  same  time ;  this  is  the  beginning  of  the 
eclipse.  Then  move  the  left  leg  on  the  ellipse  and  the  right  on  the 
path  until  they  also  mark  the  same  times,  which  give  the  ending. 
With  the  difference  of  the  semidiameters  51 "  find  in  the  same  man- 
ner the  beginning  and  ending  of  totality.  The  times  thus  measured 
are  found  to  agree  with  the  computed  times  as  near  as  a  drawing  can 
give  them.  They  are  reduced  to  mean  time  by  applying  the  equation 
of  time  (apparent  to  mean  time). 


Apparent  Time. 

Equation  of  Time.     Mean  Time. 

Eclipse  begins 

12*  5". 

—  Om  28.8              12*     2m 

Totality  begins 

131 

1   28 

Totality  ends 

137 

1   34 

Eclipse  ends 

255 

2   52 

This  projection  requires  the  utmost  exactness  in  all  its  parts,  espe- 
cially in  the  angle  of  the  moon's  path,  and  the  projection  of  that 
portion  of  the  ellipse  covered  by  the  eclipse.  The  times  of  totality 
will  be  found  especially  difficult.  All  the  quantities  that  can  be 
should  be  computed  similarly  to  those  for  the  ellipse,  so  as  to  reduce 
the  error  of  drawing.  The  drawing  for  Fig.  21  was  made  double 
the  size  of  the  figure  and  reduced  to  lessen  the  errors  of  drawing. 

164.  From  each  of  the  four  points  just  determined  on  the  ellipse, 
with  the  radius  of  the  sun's  semidiameter  953///r  describe  circles; 
and  from  the   corresponding  points  of  the  moon's  path,  with  the 
radius  of  the  moon's  semidiameter  1004"  also  describe  circles.     The 
pairs  from  the  corresponding  points  should  be  tangent  to  one  another, 
and  show  the  relative  positions  of  the  sun  and  moon  at  the  times  of 
the  four  contacts. 

165.  The  Angles  of  Position. — From  the  centre  of  the  sun  for 
beginning  and  ending  draw  lines  through  the  point  of  contact  to  the 
centre  of  the  moon.     The  angles  which  these  lines  make  with  a  line 
parallel  to  the  axis  of  Y  through  the  centre  of  the  sun's  disk  will 
mark  the  angles  of  position  in  the  disk  to  the  point  of  contact, 
measured  from  the  north  point  toward  the  celestial  east  as  positive. 
This  is  the  direction  in  which  the  sun  moves  in  the  celestial  sphere, 
from  west  to  east. 

11 


162  THEORY  OF   ECLIPSES.  165 

The  angles  as  computed  in  Art.  153  are  60°  to  the  west,  negative, 
and  119°  to  the  east,  positive;  they  are  marked  in  Fig.  21,  and  can 
be  measured  with  a  protractor.  The  first  is  negative  and  the  second 
positive,  because  the  moon  moving  the  faster,  overtakes  the  sun 
making  the  first  contact  on  the  western  limb,  and  the  last  on  the 
eastern  limb. 

For  the  total  phase,  if  we  likewise  consider  the  line  from  the  centre 
of  the  sun  through  the  point  of  contact  to  the  centre  of  the  moon,  we 
see  that  it  has  doubled  back  upon  itself,  showing  that  the  angles  of 
position  in  projection  for  the  umbral  phases  are  given  by  the  formula. 

For  Total  Eclipse        ft  ==  N  +  </>  ±  180°.  ) 
For  Annular  Eclipse  4  =  N  +  </>.  ) 

From  Art.  157  we  have  from  the  computation  for  totality: 

Q  Beginning  +  281°  53'  to  E.     Ending  +  137°  3'  to  E. 

Constant  ±180      0  ±180   8 

ft  for  Projection  +  101    53   to  E.  —    42  57   to  W. 

These  angles  are  also  shown  on  Fig.  21,  the  positive  being  meas- 
ured toward  the  lefty  and  the  negative  to  the  right,  on  account  of  the 
change  of  180°  made  above. 

If  we  consider  the  limbs  of  the  sun  on  which  the  contacts  take 
place,  we  find  them  to  be  as  follows  : 

Total  Eclipse.  Annular  Eclipse. 

First        Eclipse  begins     Partial  begins  to  W.  Partial  begins  to  W. 

Second     Umbra  begins     Totality  begins  to  E.  Annulus  formed  to  W. 

Third       Umbra  ends        Totality  ends  to  W.  Annulus  ruptured  to  E. 

Last         Eclipse  ends        Partial  ends  to  E.  Partial  ends  to  E. 

We  notice  here  that  the  change  to  the  east  limb  of  the  sun  for 
direct  vision,  the  change  of  direction,  formula  (295),  in  the  drawing, 
and  the  negative  sign  of  I  in  the  calculation,  all  mean  the  same 
thing  for  a  total  eclipse,  and  are  the  contrary  to  the  other  phases. 

166.  Partial  Eclipse. — If  the  eclipse  should  not  be  total  at  the 
given  place  on  the  ellipse,  find  the  points  of  nearest  approach  which 
mark  the  same  times  on  the  path  and  ellipse ;  the  line  joining  them 
will  be  at  right  angles  to  the  path  of  the  moon,  and  can  be  found  by 
moving  a  small  right-angled  triangle,  keeping  one  side  on  the  path, 
and  finding  where  the  other  edge  cuts  the  same  time  on  the  ellipse. 
From  these  points,  drawing  circles  as  before,  the  appearance  of  the 
sun  at  greatest  eclipse  is  shown,  and  the  distance  can  be  measured  by 


166   PREDICTION   BY   METHOD   OF   SEMIDIAMETERS.   163 

scale  on  the  cross  line ;  the  magnitude  is  the  portion  of  the  sun's 
disk  covered  by  the  moon  divided  by  the  diameter.  The  time  of 
greatest  eclipse  above  given  can  be  reduced  to  mean  time  by  applying 
the  equation  of  time. 

167.  This  method  of  finding  the  times  for  any  place  may  also  be 
applied  to  Figs*  1  and  2  in  exactly  the  same  manner  as  in  this  sec- 
tion ;  the  passage  of  the  umbral  or  penumbral  shadow  over  the  place 
will  determine  the  times ;  I  and  /x  are  given  in  decimals  of  the  sphere's 
radius  as  unity.    For  greater  accuracy,  I  or  /x  can  be  reduced  to  L  and 
Xx  by  formulae  (266)  and  (267)  for  prediction.    This  takes  account  of 
the  augmentation  of  the  moon's  semidiameter. 

168.  The  Computation. — This  method  takes  no  account  of  change 
in  the  data  by  variable  quantities,  so  that  the  error  of  a  first  approxi- 
mation is  probably  greater  than  in  the  other  method  ;  and  for  a  second 
approximation  the  times  must  be  assumed  to  the  nearest  minute  if 
possible.     Several  preliminary  reductions  are  necessary,  and  when 
the  computation  is  correctly  made,  the  results  are  correct  within  a 
fraction  of  a  second,  which  is  all  that  can  be  required. 

The  time  of  conjunction  in  right  ascension  must  be  found  at  least 
approximately,  and  two  consecutive  hours  selected  which  include  this 
time.  For  these  two  hours  the  right  ascensions,  declinations,  semi- 
diameters,  and  parallaxes  of  the  sun  and  moon  must  be  interpolated 
from  the  Nautical  Almanac  to  the  tenth  of  a  second  of  arc.  The 
moon's  right  ascension  and  declination  are  to  be  corrected  for  paral- 
lax, and  the  declination  also  corrected  for  the  convergence  of  the 
meridians  to  be  explained  presently.  The  moon's  apparent  semi- 
diameter  is  to  be  corrected  for  the  augmentation  ;  this,  however,  may 
be  omitted  in  the  first  approximations,  but  it  has  great  influence  upon 
the  times  of  the  total  phase  or  annular  eclipse. 

Maps  of  the  earth's  surface  are  usually  drawn  with  the  north  at 
the  top,  and  the  shadow  of  the  moon  in  a  solar  eclipse  will  be  shown 
as  moving  from  the  left  toward  the  right.  If  now  we  face  the  sun 
toward  the  south,  the  moon  will  be  seen  to  encroach  upon  the  sun  on 
the  right  hand,  and  to  move  from  right  to  left;  and  this  method  by 
semidiameters,  being  founded  upon  their  visual  contact,  their  relative 
positions  must  be  represented  as  they  would  be  seen  to  the  eye. 
Therefore,  in  Fig.  22,  let  Z  represent  the  centre  of  the  sun,  sup- 
posed to  be  at  rest ;  A  the  position  of  the  moon  at  the  first  hour,  and 
B  at  the  second  hour,  moving  from  right  to  left.  ZC  will  then  repre- 


164 


THEORY   OF   ECLIPSES. 


168 


sent  the  difference  of  the  right  ascensions  for  the  first  hour  and  ZD 
for  the  second  hour.  These  will  be  noted  by  x  as  usual,  and  CA 
and  BD,  the  differences  of  declinations,  by  y.  The  hourly  motions 
are  AF=  x'  and  FB  =  y'.  And  here  it  may  be  noted  that  AF  is 
supposed  to  be  a  perpendicular  to  the  meridian  YZ  at  E,  and  if  a 
parallel  of  latitude  be  drawn  through  A,  it  will  cross  the  meridian 
YZ  below  the  point  E;  the  declination  of  A  is  therefore  less  than 
that  of  E,  so  that  a  correction  must  be  applied  to  the  moon's  declina- 
tion to  give  the  point  E  correctly.  The  declination  at  B  for  the  second 

FIG.  22. 


hour  must  likewise  be  corrected,  and  these  depend  upon  the  difference 
of  right  ascension,  ZC  and  ZD.  The  first  hour  assumed  is  considered 
as  T0,  the  other  merely  gives  the  differences  of  right  ascension  and 
declination  and  the  hourly  motions. 

Angles  measured  toward  the  left  are  here  considered  as  positive, 
and  distances  toward  the  left  are  also  positive.     Let 

The  differences  of  right  ascension,         «  —  a'  =  x 
"  of  declination,  8  —  dr  =  y 

Then  let  YZB  =  M  positive  and  ZB  =  m.     Then 

m  sin  M—  x  m  cos  M  —  y 

Also  let  the  angle  which  the  line  of  the  path  makes  with  the  meridian 
N,&ndA£  =  n.     Then 
nsmN=AF=x'  n  cosN  =  FB  =  yf 


168  PREDICTION   BY  METHOD   OF   SEMIDIAMETERS.  165 

xf  and  ?/',  being  the  hourly  motions  of  x  and  y,  they  are  also  in  fig- 
ures the  differences  of  the  hourly  motions  of  the  sun  and  moon  at 
the  assumed  hours.  N9  as  in  the  former  method,  is  always  positive, 
and  never  deviates  more  than  about  30°  greater  or  less  than  90°.  In 
the  figure  it  is  less  than  90°,  and  n  is  the  motion  of  the  shadow  in 
one  hour  =  AB.  The  angle  KBZ,  or  the  supplement  of  KAZ= 
ZALj  is  M  —  N,  and  the  line  ZI9  perpendicular  to  the  path,  is  m 
sin  (M  —  N),  and  the  angles  at  P  and  §,  taken  obtuse  for  beginning 
and  acute  for  ending,  are 

m  sin  (M  —  N) 
sin  (p  =  -  -  --  - 
sf  +  a 

in  which  sf  -\-  s,  the  sum  of  the  semidiameters,  is  either  of  the  lines 

ZP  or  ZQ. 

We  also  have  PI  =  IQ  =  (sr  -f  s)  cos  <p 

and  AI=  m  cos  {M  —  N) 

Hence  PA  =  PI-  Aland  AQ  =  IQ  +  AI 

|  PA  =  (V  -f  s)  cos  0  —  m  cos  (M  —  JV) 
(.AQ=  (sf  +  s)  cos  4>  +  m  cos  (If—  N) 

and  the  time  of  describing  these  lines  is  found  by  dividing  them  by 
the  motion  in  one  hour,  which  is  n. 

PA  is  properly  negative,  which  will  result  if  <p  is  taken  as  an 
obtuse  angle  for  beginning.  These  quantities,  noted  as  T  when 
divided  by  n,  are  to  be  subtracted  from  and  added  to  the  epoch  hour 
A  to  give  the  times. 

For  a  second  approximation  two  times  being  assumed  near  the 
resulting  times,  will  each  become  T09  and  may  be  represented  in  the 
figure  by  A  and  J5,  so  that  the  final  distances  will  be 

AP=PI-AI 


For  the  total  phase  the  path  approaches  so  near  the  centre  of  the 
sun  Z  that  with  the  difference  of  the  semidiameters  a  figure  very 
similar  to  Fig.  22  is  produced  on  a  smaller  scale,  and  the  same  rea- 
soning and  formulae  are  applicable,  substituting  s'  —  s  instead  of 
2>f  -{-  s.  This  will  give  the  shadow  for  total  eclipse  negative,  as  in 
CHAUVENET'S  discussion.  ZP  and  ZQ  will  then  represent  the  differ- 
ence of  the  semidiameters  *'  —  s  instead  of  their  sum. 

In  Fig.  22  we  can  show  the  triple  angle  of  CHAUVENET'S  formula 
(275)  of  the  previous  section.  For  beginning,  in  the  two  right-angled 
triangles  IZP  and  IZA9  it  is  the  difference  of  the  angles  at  Z.  The 


166  THEORY   OF   ECLIPSES.  168 

angle  ^  at  P  is  obtuse,  and  the  acute  angle  is  180  —  ^,  and  in  the 
triangle  IZP  the  angle  at  Z  is  then  —  90  +  (p. 

In  the  triangle  IZA  M —  N  is  the  obtuse  angle  at  A,  the  acute 
angle  being  180  —  (M—  N),  and  the  angle  at  Z  is  —  90  +  (M—  N). 

Hence  AZP  is  the  difference  of  these  angles. 

Hence  AZP=  IZA  —  IZP  =  M—  N—  </>  toward  the  left  as  positive. 

For  ending  in  the  two  right-angled  triangles  PZI  and  QZI,  it  is 
the  difference  of  the  angles  at  Z. 

In  the  first  the  angle  at  Z  is  90  —  (M—N\  and  in  the  second 
90  —  (p. 

Hence  BZQ  =  IZQ  —  IZB  =  —  $  +  M  —  .IV  toward  the  left  as  positive. 

169.  Formulae  deduced  above  are  as  follows : 
Notation  : 

a!  df  Sun's  right  ascension  and  declination  corrected  for  parallax. 

a  d    Moon's  R.  A.  and  Dec.  corrected  for  Parallax  and  the  declination 

corrected  for  convergence  of  meridian. 
TT        Moon's  parallax  reduced  for  the  latitude  of  the  place.    This  value 

is  to  be  used  for  the  corrections  in  R.  A.  and  Dec. 
s        Moon's  true  semidiameter,  constant  of  irradiation  deducted,  and 

corrected  for  augmentation. 

s'       Sun's  true  semidiameter,  constant  of  irradiation  deducted. 
<pf      The  geocentric  latitude  of  the  given  place. 
x'      The  difference  of  the  hourly  motions  of  the  sun  and  moon  reduced 

to  the  arc  of  a  great  circle. 

yr       The  difference  of  the  hourly  motions  in  declination. 
&       Moon's  true  hour  angle. 

Preliminary  reductions  for  the  above  quantities. 
Moon's  true  hour  angle. 

#  =  sidereal  time  —  Moon's  true  R.  A.  )  /ooc\ 

=  9  —  a.  j 

Geocentric  latitude  of  the  given  place 

tan  <p'  =  i/l  _  e*  tan  <p  }  (<2&1) 

or  <pf  =  <p  —  the  angle  of  the  vertical,  j 

Relative  Parallax          =  x  —  *'  from  Nautical  Almanac.  (298) 

Reduced  parallax  for  the  latitude  of  the  place. 

n  —  (TT  —  TT')  x  />  for  the  given  place.  (299) 


169   PREDICTION   BY   METHOD   OF   SEMIDIAMETERS.   167 
Parallax  in  right  ascension. 

Compute  a  =  s^"  cos  ?'  (300) 

cos  8 

And  when  the  apparent  hour  angle  affected  by  parallax  is  known 

A     sin  TTi  =  a  sin  (#  -f  77)  (301) 

When  the  true  hour  angle  &  is  known. 

B     tan,I=^^^_  (302) 

1  —  a  cos  # 

a  sin  T?    ,   a2  sin2  2#   ,   a3  sin  3# 
C  *i  =  --  -----  f-  —  (303) 

sin  1"  sin  2"  sin  3" 

This  correction  always  increases  the  hour  angle  numerically. 
Parallax  in  declination. 


,  ,  ,„.. 

Compute      cot  6  =  -  -  —  -  (304) 

cos  \U 


sin  6 
When  the  apparent  declination  affected  by  parallax  is  known 

D     sin  7r2  =  c  sin  (6  —  d  +  77  )  (306) 

When  the  true  declination  is  known 


1  _  c  cos  (6  -  d) 
F  =  c  sin  (6  -<?)   ,   c2  sin  2(6  -  d)      c3  sin  3(6-*)        , 

sin  1"  sin  2"  sin  3" 

This  correction  increases  the  distance  of  the  moon  from  the  north 
pole. 

Correction  for  moon's  declination  for  convergence  of  the  meridians  : 

Ad  =  (a  —  a')2  900  sin  I"  sin  2<?  j  (^^ 

=  [7.6398]  (a  —  a')2  sin  2<5     J 

a  —  a'  is  the  difference  of  the  right  ascensions  of  the  sun  and  moon, 
and  Ad  the  correction  in  seconds  of  arc,  to  be  numerically  added, 
whether  d  is  north  or  south. 

Augmentation  of  the  moon's  semidiameter. 

When  the  moon's  apparent  altitude  is  known,  and  the  parallax  in 
declination  7T2  is  known, 

A  =  s  sin  TT  cot  (b  —  #)  —  £  s  sin2  -K  (310) 


168  THEORY   OF   ECLIPSES.  169 

b  is  the  quantity  in  equation  (304)  above  given,  and  (b  —  d)  is  used 
in  equations  (307-8).  This  quantity  is  to  be  added  to  the  moon's 
semidiameter. 

The  hourly  motions  in  right  ascension  reduced  to  the  arc  of  a 
great  circle 

xr  =  (Aa  —  Ad)  cos  5  (311) 

In  LOOMIS'  Practical  Astronomy  are  given  tables  by  which  several 
of  the  above  quantities  may  be  taken  out  at  once.  The  correction 
for  the  sun's  parallax  in  formulae  (298)  is  small,  so  that  it  is  here 
included  with  that  of  the  moon,  and  both  reduced  together.  In  the 
reduction  for  ^',  the  radius  of  the  earth  for  the  latitude  is  to  be 
used.  This  may  be  taken  from  Table  IV.  or  the  angle  of  the  verti- 
cal interpolated.  For  the  parallaxes  in  right  ascension  and  declina- 
tion several  formulae  are  given,  either  of  which  may  be  used,  according 
to  circumstances.  The  semidiameters  of  the  sun  and  moon  given  in 
the  body  of  the  Almanac  are  the  apparent  values,  which  contain  con- 
stant terms  for  irradiation,  as  explained  in  Arts.  20  and  21.  These 
constants  must  be  deducted  first,  then  the  moon's  semidiameter  cor- 
rected for  augmentation. 

Principal  formulae,  Fig.  22, 

Differences  of  R.  A.        x  =  (a  —  «')  1 
Differences  of  Dec.         y  =  (3  —  <?')  ) 

smM=x,  (313) 


Hourly  motions,    n  sin  N=  (Aa  —  Aa')  cos  8 


C314") 

.  ,  —  /oi  n\ 

sm  <p  =  ±  -  2  --  L  (315) 

s'dbs 

<     ^  g   cos  $       m  cos    M—  N 


T=T0  +  r  (317) 

Augles  and  distances  measured  from  right  to  left  are  positive.  <p  is 
to  be  taken  as  obtuse  for  beginning  with  its  cosine  negative,  and 
acute  for  ending  with  its  cosine  positive.  In  equations  (315)  and 
(316)  the  upper  sign  of  (sr  zt  s)  gives  the  times  of  a  partial  eclipse, 
and  the  lower  the  total  or  annular  eclipse.  For  a  total  eclipse  only, 
the  lower  signs  before  these  equations  are  to  be  used  to  counteract 
the  negative  of  this  factor  in  equation  (315). 


169  SHAPE  OF  THE  SHADOW  UPON  THE  EARTH.  169 

Equation  (316)  may  be  put  into  another  form  giving  the  triple 
angle.     Multiply  numerator  and  denominator  by  sin  <fi. 

sin  <f>  (sf  ±  s)  cos  0  —  m  cos  (M—N)  sin  0 
n  sin  </> 

Substitute  the  value  of  sin  <p  (sr  ±  s)  from  (314), 

m  sin  (M  —  N)  cos  0  —  m  cos  (M  —  N)  sin  <p 
n  sin  (> 


In  LOOMIS'  Astronomy  this  equation  is  given  thus, 


which  results  from  angles  being  considered  numerically,  without 
regard  to  .sign,  and  also  because  N  is  taken  as  acute  ;  =  90  ±  JV  of 
CHAUVENET'S  notation. 

In  using  these  formulae,  at  least  one  decimal  of  arc  should  be 
retained  after  making  the  reductions,  which  will  give  five  figures  for 
the  quantities  in  seconds  considered  as  linear;  and  compute  with 
five-place  logarithms  to  seconds  of  arc.  Signs  must  be  regarded  in 
the  above  formulae,  and  the  angles  of  position  are  found  as  in  the 
former  section.  In  a  partial  eclipse  the  middle  time  is  given  by  the 
second  term  of  equation  (316),  and  the  magnitude  can  be  computed 
as  in  Art.  156. 


SECTION    XX. 

SHAPE  OF  THE  SHADOW  UPON  THE  EARTH. 

170.  Shadow  Bands. — My  attention  was  first  called  to  the  subject 
of  this  section  during  the  total  eclipse  of  the  sun,  May  28,  1900, 
while  at  Newberry,  South  Carolina,  where  I  went  with  Professor 
Cleveland  Abbe  and  Professor  Frank  H.  Bigelow,  of  the  Weather 
Bureau,  to  observe  that  eclipse. 

Just  before  and  just  after  the  totality  of  a  solar  eclipse,  certain 
lines  of  light  and  shade  are  sometimes  seen  on  the  ground,  moving 
perhaps  in  a  direction  at  right  angles  to  their  length.  They  have 


170  THEORY  OF  ECLIPSES.  170 

been  recorded  of  various  widths  from  one  to  three  inches,  or  some- 
times more,  with  spaces  of  light  generally  greater,  even  two  or  three 
times  their  width ;  their  motion  variable,  generally  from  two  to  six  feet 
per  second,  occasionally  much  greater.  These  appearances  have  been 
denominated  Shadow  Bands,  and  their  origin  is  unknown,  but  has 
been  supposed  to  be  due  to  the  diffraction  of  the 'sun's  light.  More 
probably  they  are  caused  by  the  undulations  and  disturbances  of  the 
density  of  the  atmosphere  within  the  cone  of  shadow  caused  by  the 
lower  temperature  within  the  cone,  which  may  fall  some  five  or  six 
degrees,  thereby  producing  intermittent  opacity.  This  theory  has 
been  proposed  by  Professor  BIGELOW,  based  upon  the  voluminous 
observations  made  by  the  Weather  Bureau  during  this  eclipse. 

These  observations  were  made  over  a  tract  of  country  extending 
about  six  hundred  miles  on  each  side  of  the  centre  line,  and  over  the 
whole  of  this  region  a  decided  fall  of  temperature  was  recorded,  com- 
mencing at  each  station  on  the  centre  line  about  three-quarters  of  an 
hour  before  totality,  the  minimum  being  reached  just  at  totality,  but 
the  normal  temperature  not  being  reached  until  about  two  hours  after 
totality.  At  the  distant  stations  the  fall  was  not  quite  so  great  nor 
the  interval  of  low  temperature  quite  so  long.  If  the  shadow  bands 
are  caused  by  the  fall  of  temperature  as  above  noted,  it  seems  to  me 
that  they  must  exist  throughout  the  whole  of  this  region.  Within 
the  umbral  cone  there  is  not  sufficient  light  to  cast  a  shadow,  but 
they  are  seen  in  the  dim  light  just  before  and  after  totality,  and  the 
brighter  light  overpowers  the  shadow,  so  that  the  bands  can  be  seen 
only  just  before  and  just  after  totality,  when  the  light  is  dim. 

The  directions  for  observing  the  shadow  bands  are,  in  brief,  to 
fasten  a  sheet  upon  the  ground  stretched  free  from  wrinkles,  the 
sides  being  north  and  south.  When  the  bands  are  first  seen,  lay  a 
lath  or  straight  stick  along  their  length  and  let  it  remain ;  estimate 
how  many  bands  and  spaces  there  are  to  one  foot.  At  the  close  of 
totality  proceed  in  the  same  manner,  laying  another  stick  parallel  to 
their  length,  which  will  generally  be  in  quite  a  different  direction 
from  the  first.  After  the  shadow  bands  cease  to  be  seen,  measure  the 
directions  of  the  sticks  carefully  with  a  compass. 

Whatever  may  be  the  cause  of  the  shadow  bands,  since  they  are 
seen  only  around  the  shadow  of  total  eclipse,  it  is  important  in  any 
discussion  of  them  to  know  the  contour  of  the  shadow  upon  the 
ground,  or,  in  other  words,  the  direction  of  the  tangent  line  at  any 
place  where  they  may  have  been  observed.  It  is  not  known  to  the 
author  that  this  problem  has  heretofore  been  worked  out. 


171  SHAPE  OF  THE  SHADOW  UPON  THE  EARTH.  171 

171.  TJie  Ellipse  of  Shadow. — The  shadow  of  a  total  eclipse  as  it 
appears  upon  the  surface  of  the  earth,  disregarding  local  irregulari- 
ties, is  the  intersection  of  a  sphere  and  cone,  the  compression  of  the 
earth  being  here  neglected.  An  oblique  section  of  a  cone  by  a  plane 
is  an  ellipse,  but  no  plane  can  cut  an  ellipse  from  a  sphere.  The 
intersection  is  therefore  a  curve  of  double  curvature,  but  since  the 
cone  is  quite  small  compared  with  the  sphere  of  the  earth,  the  small 
portion  of  the  earth  approaches  to  a  plane,  so  that  the  curve  ap- 
proaches to  an  ellipse,  and  practically  is  an  ellipse,  as  will  be  seen 
below. 

The  minor  axis  6  of  this  ellipse  is  evidently  the  radius  of  the 
shadow  at  the  height  of  the  given  place  above  the  fundamental  plane, 
which,  for  a  point  of  the  central  line  of  shadow,  is  the  quantity  L  in 
the  computation  for  Duration,  Section  XIV.,  Arts.  116  and  117  ;  but 
being  given  there  in  parts  of  radius,  it  must  be  divided  by  the  sine 
of  one  minute  to  reduce  it  to  minutes  of  arc.  The  axis  of  the  cone 
of  shadow  during  totality  is  very  nearly  the  line  from  the  observer 
(anywhere  within  the  central  path)  to  the  sun,  and  the  radius,  L,  is 
measured  at  right  angles  to  this  axis.  The  surface  of  the  earth  at 
this  point,  being  the  horizon  of  the  observer,  is  inclined  to  the  axis 
of  the  shadow  by  the  zenith  distance  f  of  the  sun,  and  the  oblique 
projection  of  L  on  the  plane  of  the  earth's  surface  is  6  divided  by 
cos  £,  which  is  a,  the  major  axis  of  the  ellipse  of  shadow.  The  zenith 
distance  is  usually  computed  for  the  given  place  from  the  geographical 
zenith,  but  if  computed  for  the  place  of  the  centre  of  the  ellipse  for 
the  geocentric  zenith,  it  is  seen  to  be  the  quantity  cos  ft  already  made 
use  of  in  the  computation  for  the  Central  Line,  Section  XII.,  Arts. 
108  and  111,  so  that,  substituting  cos  ft  for  cos  f,  we  have  the  quan- 
tity X  used  in  the  Limiting  Curves  of  Total  Eclipse,  Section  XVI., 
Arts.  136  and  138. 

One  more  element  is  required  to  determine  completely  the  ellipse, 
and  that  is  the  direction  of  the  major  axis,  which  is  evidently  in  the 
plane  between  the  sun  and  the  zenith  of  the  place.  Assuming  this 
place  to  be  the  centre  of  the  ellipse,  this  line  on  the  earth's  surface 
is  the  azimuth  of  the  sun,  which  is  usually  measured  from  the  south 
point. 

Collecting  these  formulae  from  the  sections  above  named,  together 
with  those  for  azimuth  A  and  zenith  distance  £,  we  have  in  full : 


172  THEORY   OF   ECLIPSES.  171 

*  =  Mi  -  *>  (320) 

sin  C  sin  A  =  cos  d  sin  $  ^| 

sin  C  cos  J.  =  —  cos  y  sin  d  -f  sin  ^  cos  d  cos  #  V    (321) 

cos  C  =       sin  ^  sin  d  -f  cos  <p  cos  d  cos  &  ) 

cos  /?  =  cos  C  nearly, 
or  else  compute  sin  y9  sin  y  =  a; 

sin  0  cos  r  =  2/1  =  —  log  ft  =  9.9986  (322) 

Pi 

L=l1  —  ifcos^  (323) 

(324) 


cos     sin 
Angle  of  the  transverse  axis  from  the  north  point. 

=  180°  —  A  (325) 

Transverse  axis  a  =  ^  (326) 

Conjugate  axis    6  =  — (327) 

sin  1' 

By  these  formulae  the  axes  can  be  computed  de  novo,  d  x  y  ^  i  ^ 
being  given  in  the  eclipse  tables  of  the  Nautical  Almanac.  The 
sun's  declination  S'  should  be  used  in  place  of  d  if  the  azimuth  is 
computed  for  any  place  not  on  the  centre  line  of  the  path  of  shadow. 
They  differ  but  little,  and  one  may  be  used  for  the  other  without 
much  error,  as  none  of  the  quantities  is  required  closer  than  one  or 
perhaps  two  decimals  of  a  minute  of  arc,  using  only  four-place  loga- 
rithms, a  and  b  will  result  in  minutes  of  arc,  which  is  taken  as  the 
unit  of  measure  in  this  section.  Angles  are  measured  to  the  nearest 
minute  of  arc,  and  these  will  be  sufficiently  exact.  The  direction  of 
shadow  bands  cannot  be  measured  closer  than  one  degree  of  arc. 

If  this  section  should  be  of  use  to  astronomers,  the  quantities  L 
and  A,  the  axes  of  the  ellipse  of  shadow  together  with  cos  /9,  the  zenith 
distance  can  be  furnished  by  the  Nautical  Almanac  office,  for  they 
are  already  used  in  the  calculations,  or  they  can  be  given  for  every 
ten  minutes  in  the  special  pamphlet  usually  published  in  advance  of 
any  total  eclipse  visible  in  the  United  States. 

172.  The  Axes  and  Conjugate  Diameters. — It  remains  to  show  that 
the  above-mentioned  lines  a  and  b  are  really  the  axes  of  the  ellipse 
of  shadow.  If  the  ellipse  is  centred  upon  one  of  the  five-minute 
points,  for  which  the  centre  line  and  limits  are  given  in  the  Nautical 
Almanac,  the  line  joining  these  two  points  may  be  regarded  as  a  con- 
jugate diameter  of  the  ellipse,  since  on  the  earth's  surface  at  its  ex- 


172  SHAPE  OF  THE  SHADOW  UPON  THE  EARTH.  173 

tremity  the  limiting  line  is  tangent  to  the  shadow.  On  the  funda- 
mental plane  this  line  is  determined  by  the  quantity  §,  so  taken  as 
to  be  at  right  angles  to  the  direction  of  the  motion  of  the  shadow ; 
the  tangent  at  its  extremity  is  therefore  also  parallel  to  the  direction 
of  the  motion.  It  therefore  occurred  to  me  that  if  another  computa- 
tion were  made  using  the  formulae  for  limits,  but  taking  Q  differing 
90°  from  the  former  value,  we  should  obtain  the  other  conjugate 
diameter  of  the  circle  on  the  fundamental  plane,  the  tangent  at  the 
extremity  of  each  being  parallel  to  the  other  line.  And  in  projection 
upon  the  plane  of  the  earth's  surface,  since  parallel  lines  will  also  be 
parallel  in  projection,  these  lines  and  tangents  will  be  conjugate  diam- 
eters of  the  ellipse  of  shadow.  This  reasoning  is  found  to  be  cor- 
rect, and  the  proof  of  the  ellipse  as  well  as  of  the  axes  and  conjugate 
diameters  is  given  by  the  property  of  the  ellipse,  that  the  sum  of  the 
squares  of  the  axes  is  equal  to  the  sum  of  the  squares  upon  any  set 
of  conjugate  diameters. 

The  shadow  of  a  carriage  wheel  on  the  ground  is  a  good  though 
homely  illustration  of  this  theorem.  A  cane  placed  on  the  tire  in 
the  plane  of  the  wheel  and  perpendicular  to  one  of  the  spokes  will 
show  in  shadow  a  conjugate  diameter  if  there  is  another  spoke  par- 
allel to  the  cane.  If  the  conjugate  diameters  can  be  so  selected  that 
they  and  their  shadows  are  at  right  angles,  they  will  be  the  axes  of 
the  ellipse  of  shadow. 

In  this  example  we  might  have  shown  this  proof  of  the  point  of 
the  centre  line  and  limits  given  in  the  examples  of  Sections  XII., 
XIV.,  and  XVI. ;  but  as  the  quantities  L  and  X  will  be  needed 
for  other  problems  in  connection  with  the  Given  Place  of  Section 
XVIII.,  we  here  compute  another  point  of  the  central  line,  dura- 
tion and  limits,  at  9*  30  .  The  work  for  the  two  first  of  these  is 
given  in  brief,  but  sufficient  to  show  the  correctness  of  L  and  A, 
which  are  the  axes  of  the  ellipse  a  and  b.  The  work  for  limits  is 
given  in  full  and  headed  "  Limits  for  bf"  It  is  identically  the  com- 
putation for  the  limiting  curve  given  in  the  Nautical  Almanac.  The 
resulting  geographical  positions  are  the  ends  of  the  line  we  have 
chosen  as  a  conjugate  diameter,  6'.  In  addition  to  this,  the  last  col- 
umn gives  a  similar  computation  to  the  previous  and  computed  by  the 
same  formulae,  except  that  Q  is  increased  by  90°,  making  an  obtuse 
angle.  The  resulting  geographical  positions  of  this  latter  are  on  the 
centre  line,  and  are  also  the  two  extremities  of  the  other  conjugate 
diameter,  ar.  In  the  present  case  af  is  here  greater  than  6',  but  this 
may  not  always  be  so  if  the  axis  b  is  greater  than  a. 


174 


THEORY   OF   ECLIPSES. 


172 


The  computation  for  the  axes  and  conjugate  diameters  is  given  in 
the  following  example  : 

AXES  AND  CONJUGATE  DIAMETERS  (9A  30m). 


Axes  a  and  6. 

Conjugate  Diameters  a'  and  b'. 

Centre  Line  in  brief. 

Limits,  for  6'.    New  point,  for  a'. 

9*30 

9*  30                                    9*  30 

N.  A.  x  +0.39586  +9.57503 

(250)  A                      —1.8417                      —1.8417 

N.  A.  y  —0.28685  —9.45766 
(206)  (207)  yl                      9.45911 

Ur^l+9.6930sin($+90°)  +9.9395 

tan  y                           0.11592 
y                        +  127  26  33 

AsmJ3"=|    _|_99395                      —9.6930 
cosQ      i 

sin  y                          9.89980 

N.  A.  sin  dj     +8.9620                      +8.9620 

sin/3                           9.67523 
(208)  cos/3                           9.94490 

(251)  \  COSH=\  +8.6550                      +8.9015 
sin  Q  sin  dt  > 

tan  H                 1.2845                          0.7915 

(209)  &                   +  22°  35'  16" 

H                    87  1  36                   —80  49  9 

N.A.//!             143    11  51 

sin  H              +9.9994                      +9.9944 

6>                       120    36  35 

(251)  log  h               +9.9401                       +9.6986 

tan  ^i                      —9.32307 

#  from  Cen-  j  +64  2fi  20             _j_103  24  2_ 

(210)  tan  0                       —9.32454 

tral  w     .n  J 

<t>                     —11°  55/  18" 

sin  ($—H)        9.9553                      +9.9880 

cos  (&—H)        9.6350                          9.3652 

Duration. 

(252)  <fy  +54.589    +1.7371       —33.75  —1.5283 

(219)  N.A./!  —0.013725  —8.1375 
N.  A.  log  i               +7.6647 
log  i  cos  P                +7.6096 

(252)  log  (l)+5.494  +0.7399     —  1.693  +0.2286 
N.  A.  cos  di     +9.9982                      +0.9982 
log  (2)  —34.112  1.5329     —60.172  —1.7794 

I  —  I  A                        0.5279 

B                                0.6407 

(1)+  [—28.618                 —61.865 

log  L                       —8.2503 

(2)      J 

The  axes  a  and  b. 

The  geographical  positions. 

(21  9)  L:  cos  /?                   —8.3054 

From  the  first  column,  Limits. 

sin  1                             6.4637 

(J>                                          (0 

(326)  o  =  A             69.45—  1.8417 

(253)  North  Limit              —11°    07.7       120°  S'.O 

South  Limit              —12    49  .9      121    5  .2 

(327)  b  =  L:sin  V  61.18  —1.7866 

From  second  column.    Points  on  Centre  Line. 

(328)  Point  preceding  |    _^0  2g/  j       1190  34/  y 

(Easterly)        / 

Point  following)         -,-,     Oi    c       101     QQ   ^ 

>             J[  J,      ^J.     O           JL^J-      oo  .O 

/"\17^o*^,*l,r\ 

As  CHAUVENET  gives  a  criterion  for  the  additions  <p  +  dy  and 
co  +-  dco,  we  have  likewise  for  the  geographical  positions  resulting 
from  the  last  column  : 


The  sign  of  the  first  term  for  d<o  must  be  changed,  and 
The  Point  preceding  is  given  by  <f>  +  d<p  and  at  -\-  da> 
The  Point  following  is  given  by  <p  —  d<f>  and  to  —  da> 
For  an  Annular  Eclipse  reverse  these  conditions. 


172  SHAPE  OF  THE  SHADOW  UPON  THE  EARTH.  175 

No  further  use  is  here  made  of  the  latitudes  and  longitudes,  as  it 
is  only  the  quantities  dip  and  da)  which  we  now  require. 

173.  Proof  of  the  Ellipse.  —  We  first  require  the  length  of  the  con- 
jugate diameters.  If  we  regard  the  spherical  triangle  between  the 
point  on  the  centre  line,  the  point  of  the  limits,  and  the  pole,  we 
have  the  angle  B  at  the  centre  line  and  the  line  b  required. 

The  usual  formula?  are  rigorously 

sin  b  sin  B  —  cos  <p  sin  (u  —  a/) 

sin  b  cos  B  =  sin  <pf  cos  <p  —  cos  <?'  sin  <p  cos  (w  —  «')         (329) 

But  as  (a)  —  a/)  is  a  small  angle,  we  may  write  the  arc  for  the  sine 
and  place  the  cosine  equal  to  unity  ;  writing  the  arc  for  the  sine,  also 
for  6  and  (<p  —  <p)  in  the  second  equation,  we  have  simply 

b  sin  B  =  cos  <p  da>  )  (330") 

d  ) 


d<p  and  dco  are  here  the  differences  of  the  angle,  and  are  also  the 
quantities  d<p  and  dcu,  already  computed.  The  latter  formula?  are 
used  in  navigation,  substituting,  however,  the  mean  of  the  two  lati- 
tudes <p  and  tp'y  so  that  the  formula?  may  be  written  (Art.  127) 

b  sin  B  =  cos  %  2<pda>\  C33H 

b  cos  B  =  d<f>  } 

With  these  latter  formula?  compute  the  two  halves  of  each  conjugate 
diameter,  a!  and  bf,  then  the  proof  of  the  ellipse  is  given  by  the  equal- 
ity subsisting  between  the  sum  of  the  squares  upon  the  semiaxes 
equaling  the  sum  of  the  squares  upon  the  semiconjugate  diameter 

a2  +  bz  =  a'2  +  6'2  (332) 

And  here  it  may  be  noted  that  mathematical  accuracy  in  this 
equality  must  not  be  expected  in  the  numerical  values  for  the 
shadow  of  an  eclipse,  because  the  shadow  is  not  rigorously  an 
ellipse,  as  already  stated,  though  near  enough  for  practical  pur- 
poses and  with  approximate  formulae.  It  will  be  noticed  that  the 
two  halves  of  a  conjugate  diameter  in  this  problem  on  the  earth's 
surface  are  not  of  equal  length  because  the  degrees  of  latitude  and 
longitude  are  not  of  the  same  length  in  different  latitudes.  To  form 
the  limiting  curves,  we  lay  off  from  the  centre  line  certain  distances 
in  latitude,  then  in  longitude,  laying  off  equal  degrees  and  minutes 
for  each  limit  ;  but  toward  the  equator  a  degree  of  longitude  contains 


176  THEORY   OF   ECLIPSES.  173 

more  miles  than  a  degree  toward  the  poles,  so  that  these  distances  in 
miles  are  greater  toward  the  equator  than  those  toward  the  poles. 
The  limiting  points  resulting  are  therefore  not  at  equal  distances 
from  the  centre  line.  This  line  from  the  centre  to  the  limits  is  one 
of  our  conjugate  diameters,  and  the  two  halves  are  seen  to  be  not  of 
equal  lengths ;  and  moreover  the  three  points  do  not  lie  exactly  in  a 
straight  line.  These  circumstances  show  themselves  clearly  in  the 
computation  following.  The  effect  of  this  enters  into  our  calculations 
through  the  cosine  of  the  latitude  in  equation  (330)  given  above,  and 
in  others  of  a  similar  form. 

Besides  this,  there  is  another  cause  wholly  distinct  and  acting  in 
different  directions,  which  produces  still  greater  effect  upon  the 
length  of  all  lines,  and  that  is  the  variable  effect  of  the  sun's 
zenith  distances  at  the  two  ends  of  any  line.  The  more  obliquely 
the  shadow  falls  upon  the  earth,  the  greater  its  length.  It  is  there- 
fore greatest  at  the  beginning  and  ending  of  an  eclipse.  This  effect 
enters  into  the  calculation  of  the  limiting  curves  through  cos  /?,  the 
zenith  distance  for  a  given  place,  and  thence  to  L  Cos/9  is  here 
required,  but  is  unknown,  and  its  value  is  assumed  to  be  the  same  as 
for  the  central  line,  and  cos  ft  is  therefore  taken  from  that  computation. 
CHAUVENET  states  that  the  limiting  curves  are  not  rigorously  exact, 
and  this  is  his  principal  approximation.  We  may  see  the  amount  of 
this  approximation  roughly  in  figures  thus  :  In  the  previous  compu- 
tation for  9*  30™,  cos  ft  in  the  central  line  is  9.9449.  The  width  of 
the  shadow  path  is  about  two  degrees  at  this  time ;  a  change  of  one 
degree  on  each  side  will  change  log  cos  ft  by  0.0040,  and  this  will 
change  A  in  the  same  computation  by  about  0'.55,  making  the  semi- 
major  axis  toward  the  equator  less  and  the  other  half  toward  the  pole 
greater  by  an  amount  a  little  more.  The  total  length  would  not  be 
much  changed,  but  the  centre  would  not  lie  in  the  middle  of  the 
axis.  These  are  some  of  the  difficulties  we  have  to  contend  against 
in  regarding  this  shadow  as  an  ellipse  and  the  earth  a  plane. 

The  numerical  work  for  formulae  (331)  and  (332)  is  as  follows :  We 
compute  each  half  of  the  conjugate  diameters  because  they  are  not  of 
the  same  length. 

The  data  in  the  work  below  are  taken  from  the  previous  example. 
Signs  are  omitted,  since  it  is  only  numerical  values  that  are  required, 
and  the  four  extremities  of  the  conjugate  diameters  are  noted  by  the 
points  of  the  compass.  £  £<p  in  the  formula  is  the  mean  of  the  lati- 
tudes of  the  centre  of  the  eclipse  and  the  extremity  of  the  diameter. 
The  adjacent  semidiameters  are  combined  in  the  lower  part  of  the 


173  SHAPE  OF  THE  SHADOW  UPON  THE  EARTH.  177 


example.  The  errors  seem  large,  but  when  we  compare  the  sum  of 
the  squares  of  the  whole  diameters — that  is,  by  adding  the  two  partial 
sums  for  the  half  diameters  (which  can  be  done  in  two  ways,  as 
shown  below) — and  then  compare  with  double  the  sums  on  the  half 
axes,  the  check  becomes  almost  rigorously  exact. 

PEOOF  OF  THE  ELLIPSE. 
DATA  FROM  THE  PREVIOUS  EXAMPLE  AT  9ft  30W. 


N.  Limit. 
(331)  do              —  28/.618—  1.4566 
cos  J  20        11°  28.0    9.9912 
1.4478 
d<t>                  54.59    +1.7371 
tan£'        —27°  IV—  9.7107 
sin  cos  B'                     9.9492 
61og&'          61'.36      1.7879 

log  squares                    3.5758 

S.  Limit. 
28'.618  1.4567 
12°  22.6  9.9898 
1.4465 
54.59     1.7371 
9.7094 
9.9494 
61'.33    1.7877 

3.5754 

E.  Preceding. 
61'.86     1.7914 
12°  12.2  9.9901 
1.7815 
33.75     1.5283 
0.2532 
9.9411 
69'.25    1.8404 

3.6808 

W.  Following. 
61'.86     1.7914 
11°  38.4  9.9910 
1.7824 
33.75     1.5283 
0.2541 
9.9413 
69'.36     1.8411 

3.6822 

Sum  of  the  Squares  on  the  Axes. 
(332)  a    69.45  1.8417 

a2  4823'.9    3.6834 

b    61.18  1.7866 

62  3742'.8    3.5732 

Sum  of  \ 


squares 


8566  .7 


Squares  on  Conjugate  Diameters. 
Sums. 
N.  3765'.3     E.  4795'.!        8560'.4 
W.  4810  .8     S.  3761  .8        8572  .6 

Errors. 
—6.3 
+5.9 

Sums    8576.1          8556.9      17133.0 
Errors    -f  9  .4            —9  .8 
Double  the  squares  on  the  axes  17133'.4 

Error                                                —0  .4 

The  reason  why  the  half  sums  do  not  check  correctly  is  evidently 
because  in  the  example  we  took  the  two  halves  exactly  as  they 
resulted  in  the  previous  example,  and  the  centre  was  not  in  the 
middle  of  the  lines.  If  we  take  the  means  of  a'  and  b' ,  the  check 
is  rigorous. 

The  largest  error  on  the  half  axes,  namely,  —  9'. 8,  represents  an 
error  of  less  than  0'.04  in  a'  and  &',  for  if  they  are  each  increased 
by  this  amount,  it  is  more  than  sufficient  to  reduce  this  error.  There- 
fore, we  may  regard  all  the  lines  correct  within  about  0'.04.  The 
axes  and  their  squares  are  here  regarded  as  being  rigorously  correct. 

174.  Construction  of  the  Ellipse  Graphically. — For  most  purposes 
it  may  be  sufficient  merely  to  make  a  drawing  of  the  ellipse  of 
shadow  from  the  axes  computed  by  the  formulae  (Art.  171),  or  fur- 
nished by  the  Nautical  Almanac  office.  If  intended  for  use  .along 
an  extended  stretch  of  country,  it  may  be  necessary  to  make  several 
drawings  of  the  ellipse,  since  both  the  eccentricity  and  direction  of 

12 


178 


THEORY   OF   ECLIPSES. 


174 


the  major  axis  usually  change  greatly  during  one  eclipse.  The  eccen- 
tricity will  change  the  greatest  at  the  beginning  and  ending  upon  the 
earth,  while  the  directions  of  the  major  axis  will  change  the  more 
rapidly  the  path  passes  near  the  point  Z  of  the  earth  (Art.  38).  For 
this  purpose  the  azimuth  and  zenith  distance  of  the  sun  are  required 
at  the  place  of  the  centre  of  the  shadow  and  at  the  given  time,  and 
also  the  axes  a  =  A  and  L,  from  which  b  is  obtained. 

In  the  following  example  we  will  compute  these  quantities  for  the 
ellipse  at  the  beginning  and  ending  of  the  eclipse  given  in  the  exam- 
ple (Section  XVIII.).  For  purposes  required  in  future  articles,  the 
ellipses  are  here  computed  for  the  exact  times  of  beginning  and 
ending.  Generally,  however,  if  the  choice  is  given,  it  will  be  sim- 
pler to  centre  the  ellipses  upon  one  of  the  five-minute  points,  for 
which  the  eclipse  data  are  given  in  the  Nautical  Almanac.  Four- 
place  logarithms  for  the  azimuth  will  be  sufficient,  as  it  is  required 
only  to  the  nearest  minute  of  arc.  The  quantities  <p  and  a)  are  points 
of  the  centre  line  of  the  eclipse  interpolated  from  the  Nautical  Alma- 
nac eclipse  tables,  for  the  times  of  beginning  and  ending,  found  in 
Section  XVIII.  ;  the  other  quantities,  marked  N.  A.,  are  from  the 
eclipse  tables  in  the  Nautical  Almanac.  The  quantities  X  and  L  are 
from  the  computing  sheets  of  the  eclipse.  These,  however,  for  9A 
30m  have  been  computed  in  the  foregoing  pages  of  this  section. 

COMPUTATION  FOR  THE  ELEMENTS  OF  THE  ELLIPSE  OF  SHADOW. 


For  the  Beginning  and  Ending  at  the  Given  Place  (Section  XVIII.). 


Data. 

T.             9*  28"».56        9»  34"».04 

N.A. 

0     —  11°  38'  A  —  12°43/.2 

N.A. 

«       121      7  .2     119     8  .2 

N.A. 

/*!      142    50.2     144   12.4 

#     +21    43  .0  +25     4  .2 

sin#  +9.5682       +9.6271 

cos#  +9.9680       +9.9570 

N.A. 

sind  +8.9605       +8.9604 

N.A. 

cosd  +9.9982       +9.9982 

sin0   —9.3049       —9.3428 

cos0    +9.9910       +9.9892 

(321)  cos  (j>  sin  d 

sin  0  cos  d  cos  # 
sin  C  cos  A 
sin  C  sin  A 
A 

(321)  sin  C 

sin  0  sin  d 

cos  0  cos  d  cos  # 

cos  C 

C 


9*  28"«.56  9*  34".  04 
+8.9515  +8.9496 
—9.2711 
—9.4411 


+9.5664 


—9.2980 
—9.4589 
+9.6253 


+126    51    +124,  -7-7, 


+9.6632 
+8.2654 
+9.9572 
9.9483 
+27°  25r 


+9.7082 

+8.3032 

+9.9444 

9.9343 

30°  43/ 


Time. 

9A20 
25 
30 
35 
40 


Data  for  the  Axes  of  Ellipse  from  Nautical  Almanac  Computations, 
log  A. 


.8417 

.8522 

1.8641 


88 
105 


a  Numbers. 
66.97  , 
68.06  T" 
69.45 
71.14  , 
73.13  T 


logi. 
8.2551 

8.2503 


log  (L :  sin  1'). 

1.7914 

.7892 


199 


.7866 
.7837 
8.2440         1.7803 


26 
29 
—  34 


6  Numbers. 
61.86 
61.55" 


61.18 
60.76 
60.30 


37 
42 
—  46 


174  SHAPE  OF  THE  SHADOW  UPON  THE  EARTH.  179 

(326)  (327)  Axes  for  the  Times  of  Beginning  and  Ending. 

log  A.  log  a2.  a.       log  L :  sin  1'.       log  62.  6. 

Beginning    9A  28TO.56        1.8392        3.6784        68'.06        1.7873        3.5746  6K28 

Ending         9  34  .04        1.8501        3.7002        70 .81        1.7843        3.5686  60 .86 

From  the  above  data  the  ellipse  can  be  constructed.  The  azimuth 
gives  the  direction  of  the  major  axis  from  the  north  point,  while  a 
and  b  are  the  axes.  The  best  method  for  construction  is  that  by  con- 
centric circles  on  the  axes  described  in  Art.  162.  In  the  present 
example,  the  centres  must  be  located  on  the  centre  line  of  the  eclipse 
by  their  latitudes  and  longitudes.  This  is  shown  in  Fig.  23,  Plate 
IX.,  drawn  on  a  large  scale  of  20  minutes  of  arc  to  one  inch  and 
reduced  one-half;  that  is,  40  minutes  of. arc  to  an  inch  in  the 
printed  plate.  The  convergence  of  the  meridian  shows  itself  in 
this  figure,  wfyich  is  taken  into  account  by  shortening  the  degrees 
of  longitude  for  the  latitude  of  the  given  place  11°  54,'  making  one 
degree  of  longitude  equal  to  58 '.6  9  of  latitude  and  drawing  the 
meridian  lines  parallel.  This  avoids  curved  lines  for  the  meridians, 
but  it  also  occasions  a  small  discrepancy  of  measurement  throughout 
the  drawing. 

In  fact,  in  whatever  way  the  earth's  curved  surface  maybe  repre- 
sented in  a  drawing,  discrepancies  arise.  Various  expedients  have 
been  devised  to  obviate  them,  giving  rise  to  the  several  projections — 
stereographic,  globular,  mercators,  polyconic,  etc.  Each  has  its 
advantages  for  the  purpose  intended,  but  likewise  each  has  its 
defects. 

The  curvature  of  the  centre  line  is  apparent  in  Fig.  23,  but  it  has 
been  drawn  as  a  straight  line  between  the  five-minute  points,  which 
are  marked  by  small  circles.  It  is  thus  seen  that  we  have  mechan- 
ical as  well  as  mathematical  difficulties  confronting  us  in  this  section, 

175.  Computation  for  the  Tangent  to  the  Ellipse  at  any  Point. — Be- 
sides the  graphic  method,  we  will  now  derive  a  series  of  formulae  by 
which  the  tangent  to  the  ellipse  at  any  point  may  be  computed.  The 
given  point  must  of  course  lie  on  the  curve  of  the  ellipse,  and  its 
latitude  and  longitude  must  be  accurately  known.  Then  the  follow- 
ing data  must  be  taken  from  the  eclipse  tables  in  the  Nautical  Alma- 
nac,  viz. :  The  latitudes  and  longitudes  of  several  points  in  the  centre 
line,  from  which  to  interpolate  the  centres  of  the  ellipses ;  sin  c?,  cos  d, 
and  //u  for  computing  the  azimuth  (Art.  171);  the  logarithms  of  the 
axes  of  the  ellipse  furnished  by  the  Nautical  Almanac  office  or 
computed  by  Art.  171. 


180  THEOKY   OF   ECLIPSES.  175 

Points  whose  latitudes  and  longitudes  are  accurately  known  are 
marked  on  Plate  IX.,  Fig.  23,  by  small  circles ;  the  centres  of  the 
ellipses  are  accurately  known  by  interpolation.  Between  any  three 
of  these  adjacent  points  the  surface  of  the  earth  is  considered  as  a 
plane  triangle  passing  through  the  points,  and  we  may  first  show 
what  error  is  made  in  this  assumption.  The  longest  line  in  the  fol- 
lowing computation  is  the  semi-axis  of  the  ellipses.  Assuming  a 
mean  value  of  the  two,  we  have  70'. 00  in  round  numbers.  The  chord 
of  this  arc  is  twice  the  sine  of  half  the  angle,  or  in  natural  numbers 
0.0203618.  Dividing  this  by  sin  V,  the  result  for  the  chord  is 
69.983,  an  error  of  0'.017  between  this  and  the  arc.  The  rise  of 
the  arc  above  the  middle  of  the  chord  is  1  —  cos  of  half  the  angle, 
or  0.0000518  in  parts  of  radius,  and  dividing  by  sin  1',  we  have 
OM78.  The  former  is  almost  insignificant,  affecting  only  the  second 
decimal,  while  the  latter  has  but  little  effect  upon  the  lengths  of 
lines,  and  is  the  elevation  only  of  a  high  hill  and  less  than  the  devia- 
tions of  the  elevations  in  a  mountainous  district. 

In  Fig.  23  are  shown  the  centre  line  and  limits  of  the  total  Eclipse 
of  Sept.  9,  1 904,  and  the  ellipses  of  shadow  at  the  instant  of  begin- 
ning and  ending  of  the  total  phase  at  the  given  place  P  assumed  in 
Section  XVIII. 

Angles  are  here  measured  from  the  north  of  the  meridian  of  the 
centres  toward  the  left  as  positive ;  and  although  the  ellipse  itself 
moves  from  left  to  right  across  the  position  of  the  given  place,  the 
apparent  motion  of  the  given  place,  analytically,  is  from  right  to  left 
through  the  ellipse,  which  is  the  positive  direction  according  to  this 
system. 

In  order  that  the  angle  of  the  path  with  the  meridian  may  be 
measured  accordingly,  it  is  necessary  to  change  the  signs  of  both  dtp 
and  da),  so  that  in  the  next  following  formula  the  second  members 
are  given  the  negative  sign;  the  angle,  therefore,  differs  180°  from 
that  used  in  prediction ;  the  two  angles  are  of  the  same  nature,  but 
cannot  be  directly  compared,  since  one  is  measured  from  the  axes 
of  coordinates  and  the  other  from  a  meridian.  We,  therefore,  have — 

Angle  of  the  centre  line  of  the  path  of  shadow, 

n  sin  N .  =  —  datf  cos  %2<p\ 
n  cos  N  =  —  d<p  ) 

d<p  and  da)  are  here  interpolated  from  the  differences  of  the  centre 
line  for  the  times  of  beginning  and  ending  of  the  eclipse.  For  ex- 
ample, 1°  44'.3  is  regarded  as  the  motion  of  longitude  for  the  middle 


175  SHAPE  OF  THE  SHADOW  UPON  THE  EARTH.  181 

interval  between  9*  25m  and  9A  30m—  that  is,  at  9A  27W.5.  The  time 
of  beginning  is  at  971  28^.56,  which  is  lm.06  later,  and  this  is  a  frac- 
tion of  5  minutes  (the  interval  for  which  co  is  given).  Therefore, 
the  factor  for  interpolating  is  -f-  0.212;  and  this,  multiplied  by  5.3, 
the  mean  of  the  second  differences  of  the  motion,  gives  -f-  1.10,  which 
added  to  104m.3  gives  105W.40,  the  required  value  of  the  motion  at 
the  exact  time  of  beginning.  JJty  is  to  be  understood  as  the  mean 
of  the  two  latitudes  at  the  five-minute  points.  This  gives  the  correct 
angle  of  the  path  at  the  time  of  beginning,  for  the  curvature  of  the 
path  shows  itself  in  this  figure,  as  well  as  in  the  example  following. 
Angle  between  the  path  and  Axis  of  the  Ellipse  : 
The  azimuth  being  always  measured  from  the  south  end  of  a  mer- 
idian, the  angle  from  the  north  will  be  180  —  A  ;  and  the  angle  e 
between  the  path  and  transverse  axis  is  given  by  the  expression 

4)  (334) 


which  is  taken  as  positive  in  the  position  of  Fig.  23. 
Centres  of  the  Ellipse  : 

Given  by  their  latitudes  and  longitudes  by  interpolating  the  ) 
central  line  for  the  given  times.  ) 

We  know  that  these  must  be  the  centres  of  the  shadow  when  the  ellipse 
touches  and  leaves  the  given  place.  In  the  computation  following  the 
latitude  and  longitude  of  the  given  place  are  placed  between  those  of 
the  centres  of  the  ellipse  for  beginning  and  ending,  from  which  the 
correct  signs  of  the  differences  can  be  gotten  for  the  next  equation. 

Find  the  distances  r  of  the  given  place  from  the  two  centres  for 
beginning  and  ending  by  the  formula  : 

Regard  the  signs  by  taking  in  all  cases  for  A<p  and  Aa>,  the  )     /-oog\ 
signs  resulting  from  given  place  minus  the  centre.  ) 

r  sin  R  =  Au>  cos  \2<  )  /oo7\ 


r  is  the  required  distance  and  R  the  angle  which  this  line  makes 
with  the  meridian,  which  will  result  as  negative  for  beginning  and 
positive  for  ending. 

Call  o  the  angle  which  this  line  r  makes  with  the  transverse  axis 
of  the  ellipse,  measured  from  the  west  end  of  the  axis.  It  will  be 
obtuse  for  beginning  and  acute  for  ending. 

With  o  and  r  it  is  seen  that  the  given  place  is  now  referred  to  the 
centre  and  axes  of  the  ellipse  by  polar  coordinates  ;  and  we  have  the 


182  THEORY   OF   ECLIPSES.  175 

following  relations,  taking  i>  as  obtuse  for  beginning  and  acute  for 
ending. 

u  =  N  —  R  —  e.  (338) 

With  this,  working  backward  from  the  usual  formulae  for  the  angle 
of  the  vertical,  we  get  the  angle  w  of  the  normal  to  the  ellipse  at  the 
point  P,  which  is  usually  in  the  same  quadrant  with  i>. 

a? 
tanw=—  tan  w,  (339) 

a  and  b  being  the  axes  of  the  ellipse  previously  found.     And  the 
angle  which  the  normal  makes  with  the  path  of  the  shadow  is 

p  =  w  +  e  (340) 

The  angle  between  the  tangent  and  the  path  is  evidently  the  com- 
plement of  this  angle  (since  the  angle  at  P  is  a  right  angle),  but 
measured  in  the  position  of  this  section,  it  becomes 

Angle  between  the  tangent  and  path  =  p  —  90° 

Finally,  the  angle  between  the  tangent  and  the  meridian  from  the 
north  point  toward  the  left  is 

For  beginning,  t  =  (p  —  90)  —  N  ) 

For  ending,        t  =  (p  —  90)  +  (180  —  JV)  3 

These  latter  may  be  simplified,  but  they  are  convenient  as  they 
stand. 

Combining  the  last  three  equations,  we  have  generally  for  both 
beginning  and  ending  the  angle  of  the  tangent  line  with  the  merid- 
ian of  the  centre 

t=  w  +  e=p9Q  —  N  (342) 

The  formulae  are  sufficiently  simple,  and  they  are  illustrated  in 
the  following  example,  which  doubtless  requires  no  further  explana- 
tion. The  given  place  is  that  of  Section  XVIII.,  and  other  data 
are  from  the  previous  examples  of  this  section.  The  several  lines 
and  angles  here  computed  can  be  measured  in  Fig.  23. 

COMPUTATION  FOB  THE  TANGENT  TO  THE  ELLIPSE. 

Data  from  the  Nautical  Almanac  and  Previous  Examples. 

N.A.         Central  line   9*  20*  0—10°    (K.2     ,7n  w  124°   (K.3    ,  OQ  A 

25  10   57  .2~~i?™— 1.1  122   20  .9     ,  ffJ— IJ 

30  11    55 .3    gJ'J     1-2  120   36 .6    !  ™    5.7 

35  12   54  .6    Sfi—1.6  118   46  .6    }  ™ "— 7.3 

40  -13   55  .4~60'8  116  49.3- 

(Art.  151)  The  times  Beginning    9»  28^.56       Ending    9*  34™04 

(Art.  174)  Azimuth  126°.51'  124°  W 

(Art.  174)  Zenith  distance  and  cosine    27°  25/         9.9483         30°  43'        9.9343 


175  SHAPE  OF  THE  SHADOW  UPON  THE  EARTH.  183 


Angle  of  the  Centre  Line  of  the  Path. 


(333)          (Change  signs)  du 


+105.42       +9.0229      +112.00      +2.0492 


cos (1:2)20 —11  26.3    +9.9913      —12  25.0  +9.9897 


n  sin  N                                                   +2.0142 

+2.0389 

ncosN             d0                     58.34      +1.7660 

+59.72 

+1.7761 

tanJV                                 0.2482 

0.2628 

N                                                          +60°  33' 

+61°  22' 

sin  N                                 9.9399 

9.9433 

n  log  n                                     118.66          2.0743 

124.62 

2.0956 

Angle  between  the  Path  and  Major  Axis. 

(Art.  174) 

(180  —  4)                               +53    9 

+55  43 

(334) 

e  =  N—  (180  —  A)                               +  7  24 

+  5  39 

Centres  of  the  Ellipses. 

Factors  for  interpolation        +0.712     +9.8525 

+0.808 

+9.9074 

A?                 —58.1        —1.7642 

—59.3 

—1.7731 

—41.37      —1.6167 

—47.92 

—1.6805 

Correction  1 

for  A,    1  +°-12 

+0.10 

0                                       —11  38.45                    —12  43.12 

(335) 

Au              —104.3        —2.0183 

—110.0 

—2.0414 

7f!^7  —  IVM.W     —1.8700 

—  f  28.88 

—1.9488 

Cor.  for  A2      +0.54 

+0.50 

+121  7.17 

119  8.22 

Centre  of  ellipses  at  beginning  <j>  —  11  38.4  -.re 

[ 

+121  7.2 

Position  of  the  given  place            —  11  54.0  ^g'j 

+120  0.0 

Centres  of  ellipses  at  ending         —  12  43.1 

+119  8.2 

Distances  of  Given  Place  from  Centres  of 

Ellipses. 

(336) 

Place—  Centre  du                    —67.2    —1.8274 

+51.8 

+1.7143 

(337) 

cos  (1:2)20  —1146.2     +9.9908 

—12  18.6 

9.9899 

r  sin  R                           -  1.8182 

1.7042 

(336) 

Place—  Centre  r  cos  R       A0  —  15.6    —1.1931 

49.1 

+1.6911 

(337) 

tan  R                                 0.6251 

0.0131 

R                                 —103  20 

+45  52 

sin  R                                 9.9881 

9.8560 

r  log  r                  67.62      1.8301 

70.50 

1.8482 

(338) 


Given  Place  referred  to  Centres  and  Axes  of  Ellipses. 

N                                  +60°  33'  +61°  22' 

R                                 —103  20  +45  52 

N— R                        +163  53  +15  30 

e                                      +7  24  +5  39 

v  =  N—  R  —  e           +156  29  +9  51 


(339) 


The  Normal  to  the  Ellipse. 

tan  v  +9.6386 

(Art.  174,  Data)  log  (a2 :  6s)  +0.1038 

tan  w  +9.7424 

Angle  between  the  normal  and  axis  w  +151°  4' 


+9.2396 
+0.1316 
+9.3712 
+13°  14' 


184  THEORY   OF   ECLIPSES.  175 

Angle  of  the  Tangent  with  the  Meridian  of  the  Centres. 

(340)                                  e                                     -f-7°24'  +  5°  39' 

w  +  e                           +158    28  +18  53 

(342)                                   Constant                        —90     0  —90  0 

+68    28  —71  7 

— N                             —60    33  180  — jy +118  38 

Angle  of  the  tangent  t                            +7    55  +47  31 

176.  Check  upon  the  Foregoing  Formulae. — This  consists  in  com- 
paring the  length  of  the  line  above  noted  as  r  on  the  earth's  surface 
with  that  called  L  on  the  plane  of  the  place  P,  Section  XVIII.,  on 
the  prediction ;  this  is  done  by  projecting  the  former  line  on  the 
plane  of  the  place.  They  are  the  same,  being  the  diameter  of  the 
shadow. 

The  lines  of  greatest  and  least  curvature  of  a  surface,  as  we  read 
in  the  calculus,  lie  at  right  angles  to  one  another.  Likewise  in  an 
inclined  plane  the  lines  of  greatest  and  least  declivity  lie  at  right 
angles ;  the  latter  is  evidently  a  horizontal  line  of  the  plane,  while 
the  former  is  made  use  of  in  the  special  method  of  projection  em- 
ployed by  military  engineers  to  determine  the  positions  of  planes. 
The  centres  of  the  ellipse  and  the  position  of  the  given  place,  P,  in 
Fig.  23,  are  perpendicularly  over  these  same  quantities  on  the  plane 
of  the  place  used  in  prediction.  Fig.  21,  Plate  VIII.,  illustrates 
this.  The  major  axes  of  the  ellipses  are  lines  of  greatest  declivity 
in  their  planes,  but  the  lines  r  are  not,  and  we  therefore  do  not  know 
the  inclination  of  this  latter  line  to  the  fundamental  plane ;  but  by 
means  of  the  line  of  greatest  declivity,  we  can  project  it  thus  (Fig. 
23) :  The  projection  of  r  on  the  line  of  greatest  declivity,  the  major 
axis  is  r  cos  v ;  the  projection  of  this  on  the  plane  of  the  place  is 
r  cos  o  cos  f,  £  being  the  inclination  of  this  line  to  the  fundamental 
plane  as  well  as  the  zenith  distances  of  the  sun.  Now  project  this 
back  upon  the  line  L  through  the  angle  ulf  and  as  r  cos  u  is  the  base 
of  a  right-angled  triangle,  of  which  r  is  the  hypothenuse,  so  on  the 
fundamental  plane  r  cos  i>  cos  f  is  also  the  base  of  a  triangle,  of 
which  the  hypothenuse  is 

r  cos  o  cos  C  ,0  A  ON 

r  = =  r  cos  o  cos  C  sec  o1  C^4o) 

COS  01 

But  t>!  is  not  known,  and  to  find  it  we  must  project  the  angle  i>  in 

this  manner : 

sin  u 

tan  o  = 

cosw 

in  which  cos  u  is  a  line  of  greatest  declivity,  and  its  projection  on  the 


176  SHAPE  OF  THE  SHADOW  UPON  THE  EARTH.  185 


lower  plane  is  cos  o  cos  £ ;  but  sin  w,  being  a  horizontal  of  the 
plane,  projects  into  its  own  length  unchanged,  and  the  projection  of 
this  angle  o  on  the  lower  plane  will  then  be 


tan  ol  = 


sm  L> 


cos  o  cos  t 


tan  o 
cosC 


(344) 


These  formulae  may  be  used  as  they  stand,  or  combining  them,  we 

have 

r  sin  o 


sin 


(345) 


In  the  example  below,  the  data,  taken  from  the  previous  section, 
are  computed  for  the  times  9A  28.56  and  9A  34.04,  whereas  L  in  pre- 
diction, Art.  151,  is  computed  for  T0  =  9*  31m.O,  we  must  therefore 
interpolate  rx  for  the  interval  to  T0,  which  is  +0.445,  and  then 
compare. 

L,  however,  is  given  in  prediction  in  parts  of  radius ;  reduced  to 
minutes  we  have  as  the  quantity  for  comparison  in  minutes 

L 


sin  1' 


(346) 


In  the  example  below  the  comparison  results  very  closely,  verifying 
not  only  the  numerical  computations,  but  also  the  accuracy  of  the 
formulae.  For  tan  y,  cos  f,  sin  y,  r9  etc.,  see  examples,  Arts.  174,  175. 


EXAMPLE,  CHECK  UPON  THE  ELLIPSE  OF  SHADOW. 


Art.  175 
(344) tan  v 
cos  C 
tan  v 


156°  29' 


67'.62 


(345) r 

sin  v 

cosec  vl 

61.31 

Difference 
Interpolation  0.445  X  0.38  —  .17 

r,  at  T0 


9*  28.56 

9ft  34m  04 

9.6386 

9°51/ 

9.2396 

9.9483 

9.9343 

9.6903 

9.3053 

1.8301 

70'.50 

1.8482 

9.6009 

9.2332 

0.3565 

0.7034 

1.7875 

60.93 

1.7848 

—  0.38 

-.17 

.1.14 

(346) log  L 
sin  V 
in  arc 


8.2498 
6.4637 
1.7861 


L  in  arc        61'.11 


Error  O'.OS 


177.  Width  of  the  Shadow  Path.— This,  so  far  as  the  author  is 
aware,  has  not  before  been  made  a  subject  for  computation ;  it  is 
however  very  easily  obtained  from  data  on  the  foregoing  pages.  If 
the  line  between  any  one  of  the  computed  points  of  the  central  line 
and  the  limiting  point  be  projected  upon  the  perpendicular  to  the 


186  THEOKY  OF   ECLIPSES.  177 

path,  it  will  give  the  width.  This  line  is  already  computed  in  Art. 
172,  and  is  one  of  the  conjugate  diameters  bf  of  the  ellipse.  The 
normal  to  the  path  makes  the  angle  N  ±  90°  with  the  meridian, 
while  the  angle  of  this  line  is  E  already  computed,  whence  the  angle 
between  the  line  and  the  normal  is  (N  ±  90°)  ~  B.  Signs  may  be 
disregarded,  since  it  is  only  the  numerical  values  we  require,  and  the 
accent  may  also  be  omitted,  so  that  b  sin  (N — E)  gives  the  half 
width  of  the  shadow  path  in  minutes  of  arc. 

According  to  Col.  A.  R.  CLARKE  the  length  of  one  degree  of  the 
meridian  is  as  follows  :  * 

At  the  equator  110  567.2  metres. 

At  30°  of  latitude  110  848.5      " 

At  60°  "        "  111  414.5      " 

One  metre  =  3  280  869.33  feet. 

This  must  be  divided  by  1609.35,  the  number  of  metres  in  one 
statute  mile ;  the  value  of  b  must  be  divided  by  60  to  reduce  it  to 
degrees,  and  the  result  must  be  doubled  to  give  the  whole  width  of 
the  shadow  path.  Collecting  these  constants  and  giving  them  by 
their  logarithms  we  have  the  formulae : 

Angle  of  the  shadow  path  with  the  meridian,  computing  between 
two  conservative  points  of  the  central  line  (333). 

n  sin  N  =  Au>  cos  \2<p  ) 
n  cos  N=  A<p  ) 

Length  of  the  line  b  between  the  centre  and  either  limit. 

b  sin  B  =  Aw  cos  \Zy  |  /Q/IQ-N 

bCOS£=^  j 

Width  of  the  shadow  path  in  statute  miles. 

At  the  equator  W=  [0.35985]  b  sin  (N~B)  (349) 

Constant  for  30°  [0.36096]  ;  —  for  60°  [0.36317]. 

178.  Velocity  of  the  Shadow. — The  quantity  n  in  formula  (347), 
or  333,  is  the  space  passed  over  by  the  shadow  in  five  minutes, 
that  being  the  interval  for  which  the  points  of  the  centre  line  and 
limits  are  given  in  the  Nautical  Almanac.  To  reduce  this  from  min- 
utes, in  which  the  formulae  give  it,  to  statute  miles,  we  require  the 
same  constants  as  used  for  the  width  of  the  shadow  path,  except  the 
multiplication  by  2.  Hence  we  have  velocity  of  the  shadow  in  five 
minutes,  in  statute  miles. 

*  Tables  for  Polyconic  Projection  based  upon  Clarke's  Reference  Spheroid  of  1866. 
Special  Publication  No.  5  of  the  U.  S.  Coast  Survey,  1900. 


178  SHAPE  OF  THE  SHADOW  UPON  THE  EARTH.  187 

At  the  equator  F=  [0.05882]  n  (350) 

Constant  for  30°  of  latitude  [0.05993]  ;  -  for  60°  [0.06214]. 

179.  Computing  these  two  quantities  for  9A  30m,  for  which  we 
have  already  computed  the  distance  6',  in  Art.  172,  we  interpolate 
^and  log  n  from  Art.  175  for  this  time,  and  have  as  follows  : 

WIDTH  OF  THE  SHADOW  PATH  AND  VELOCITY  OF  THE  SHADOW. 

Art.  173  Example        At  9*  30™  B'—  27°  11'  log  6'          1.7879 

"    175  "  Interpolated  N    +  60  46  log  n  2.0817 

(349)  Equation          .N~E  87°  57/    (350)  log  n  2.0817 


sin  (N~B)  9.9997 

log  b'  1.7879 

Constant  interpolated  0.3602 

Width  in  miles  140.5  2.1478 


Constant  0.0592 

Velocity  in  5m          138.3        2.1409 

Velocity  per  hour,  1659.6  miles. 


180.  Rigorous  Computation  of  the  Ellipse  of  Shadow. — Professor 
CHAUVENET  has  pointed  out  in  Art.  320  of  his  chapter  on  Eclipses 
that  the  limits  of  total  or  annular  eclipse  may  be  computed  rigorously 
by  using  the  formulae  for  outline,  substituting  ^  for  umbra  instead 
of  I  for  penumbra.     The  ellipse  itself  as  well  as  the  limits  may  be 
constructed  in  this  manner  entirely  rigorously,  but  the  repetitions 
and  approximations  would  render  the  computation  very  laborious. 

The  expedient  resorted  to  in  Art.  172  of  this  section  suggests  a 
method  of  computing  any  number  of  points  of  an  ellipse  with  com- 
paratively little  labor;  and  the  result,  while  not  rigorously  exact,  will 
be  of  the  same  degree  of  accuracy  as  the  limiting  curves  now  printed 
in  the  Nautical  Almanac.  In  Art.  172  we  changed  the  value  of  Q 
90°  to  obtain  the  second  conjugate  diameter.  We  may  likewise 
compute  the  whole  ellipse  round  the  central  point,  by  assigning  to  Q 
successive  values  as  close  together  as  may  be  necessary  in  the  general 
formulae  of  Section  XVI.  A  great  advantage  of  this  method  is  that 
the  quantities  $,  L,  tan  <pl9  <f>,  co,  fjtl9  etc.,  from  the  central  curve  and 
duration,  are  constant  for  all  the  points  of  the  ellipse. 

181.  Circular  Shadow. — If  the  shadow  of  a  total  eclipse  can  be 
regarded  as  a  circle,  the  preceding  problem  is  very  greatly  simpli- 
fied.    In  Fig.  24  let  the  larger  circle  represent  the  moon  and  the 
smaller  the  sun,  as  in  the  method  by  semidiameters  (Fig.  21,  Plate 
VIII.).       If  the  observer  is  on    the  northern  limit  of  the   path, 
he  would  see  the  limbs  tangent  in  a  line  perpendicular  to  the  path 
and  on  their  northern  limbs.      If  on  the  southern  limit,  similarly 


188 


THEORY  OF  ECLIPSES. 


tangent  on  their  southern  limbs.     In  the  figure  the  totality  is  com- 
mencing.    The  point  of  con- 
FIQ.  24. 


tact  t  is  always  in  the  line 
joining  the  centres  of  the  sun 
and  moon.  Let  tA  (Fig.  24) 
be  the  moon's  radius,  R}  and 
tc  that  of  the  sun,  r.  Let  tp 
be  the  angle  which  the  line 
tA  makes  with  the  centre 
line,  and  a,  the  distance  of 
the  observer,  c,  from  the 
centre  line,  then  from  the 
figure — 
R  sin  <p  —  r  sin  <p  +  a 

or  sin  <p  = (351) 

R  —r 

When  the  observer  is  on  the  northern  limit,  the  line  tA,  as  above 
remarked,  is  perpendicular  to  the  path,  sin  <p  =  1,  and 

a  =  R  -  r  (352) 

which  is  the  half  width  of  the  path. 

If  the  observer  is  on  the  centre  line,  sin  <p  =  0  and  a  =  0. 
The  value  of  a  is  always  known,  being  the  position  of  the  observer ; 
R  and  r  are  also  known  and  their  difference.  If,  therefore,  a  small 
circle  be  drawn  round  the  point  A  with  the  radius  R  —  r  and  a  line 
drawn  through  it  at  the  distance  a  from  the  centre  line  and  parallel 
thereto,  the  points  where  this  line  cuts  the  circle  will  give  the  direc- 
tions of  the  two  tangents  to  the  points  of  contact  of  the  shadow  at 
beginning  and  ending. 

By  assuming  successive  values  for  a,  0.1,  0.2,  0.3,  etc.,  for  sin  (p, 
these  values  will  be  the  ratios  of  the  distance 
of  the  point  a  from  the  centre  line ;  and  con- 
sidering these  also  as  sin  y>,  the  angle  <p  is 
found  as  in  the  subjoined  table,  and  the  angle 
of  the  tangent  will  then  be  the  complement  of 
the  angle  tp. 

Angle  of  the  tangent  =  90  —  <f> 

When  the  point  c  is  south  of  the  centre 
line,  a  is  negative  and  <p  likewise  becomes  a 
negative  angle.  For  ending,  <p  will  be  greater 
than  90°. 


a 

<*> 

R  —  r 

0.0 

0° 

0' 

0.1 

5 

44 

0.2 

11 

32 

0.3 

17 

27 

0.4 

23 

35 

0.5 

30 

0 

0.6 

36 

52 

0.7 

44 

25 

0.8 

53 

8 

0.9 

64 

9 

1.0 

90 

0 

181  THE  CONSTANT  k.  189 

This  simple  solution  occurred  to  the  author  in  connection  with  the 
total  eclipse  of  May  28, 1900.*  The  elliptical  shadow  was  not  worked 
out  until  nearly  two  years  later,  when  the  present  work  was  taken  up. 

The  reader  may  perhaps  ask  why  it  is  that  in  Fig.  20,  in  which 
the  shadows  were  drawn  as  circles,  the  figure  should  give  times  and 
duration,  etc.,  so  closely  to  the  finally  computed  values,  when  we  have 
just  shown  that  these  curves  should  have  been  ellipses.  The  reason 
is  that  in  Fig.  20  we  simply  took  proportionate  parts,  not  absolute 
values ;  and  from  the  relations  that  a  circle  bears  to  an  ellipse,  the 
proportions  are  the  same  in  both.  The  ellipse  on  the  earth  projects 
into  a  circle  on  the  fundamental  plane,  and  the  proportions  of  parts 
and  lines  are  preserved,  as  we  have  shown  in  Art.  176. 


SECTION   XXI. 

THE  CONSTANT  k  IN  ECLIPSES,  AND  OCCULTATIONS  AND  OTHER 

DIFFEEENCES. 

182.  THIS  constant  is  the  ratio  of  the  moon's  radius  to  that  of  the 
equatorial  radius  of  the  earth,  and  this  latter  being  the  moon's  paral- 
lax, the  well-known  expression  results. 

k  =  —  (353) 

7T 

Professor  CHAUVENET,  in  his  chapter  on  Eclipses,  Art.  293,  p. 
448,  vol.  i.,  f  gives  as  the  value  for  eclipses  BURCKHARDT'S 

k  =  0.27227  (354) 

To  which  value  CHATJVENET  appends  a  foot-note,  as  follows :  "  The 
value  of  k  here  adopted  is  precisely  that  which  the  more  recent 
investigations  of  OUDEMANS  (Astron.  Nach.,  vol.  li.,  p.  30)  gives  for 
eclipses  of  the  sun.  For  occultations,  a  slightly  increased  value 
seems  to  be  required." 

In  this  same  chapter,  Art.  341,  p.  551,  in  treating  of  occultations 
he  gives 

log  k  =  9.435000     [In  numbers  0.272271]  (355) 

which  is  the  same  as  the  previous  value,  but  he  appends  a  foot-note 
as  follows : 

"According  to  OUDEMANS  (Astron.  Nach.,  vol.  li.,  p.  30),  we 
should  use  for  occultations  k  =  0.27264,  or  log  k  =  9.435590,  which 

*  See  Eclipse  Meteorology  and  Allied  Problems,  1902,  by  Professor  FRANK  H.  BIQE- 

,  Department  of  Agriculture,  Weather  Bureau  Bulletin  267,  pp.  55,  56. 
f  Manual  of  Spherical  and  Practical  Astronomy,     Two  vols.,  Lippincott  &  Co.,  1863. 


190  THEORY   OF   ECLIPSES.  182 

amounts  to  taking  the  moon's  apparent  semidiameter  about  1".25 
greater  in  occultations  than  in  solar  eclipses.  But  it  is  only  for  the 
reduction  of  isolated  observations  that  we  need  an  exact  value,  since, 
when  we  have  a  number  of  observations,  the  correction  of  whatever 
value  of  k  we  may  use  will  be  obtained  by  the  solution  of  our  equa- 
tions of  condition." 

And  here  the  question  naturally  arises,  Why  should  there  be  this 
difference  in  this  constant?  CHAUVENET  gives  no  reason,  nor  is 

any  generally  given  in  the  astrono- 

FlG>  25*  mies,  but  it  is  usually  understood 

to  be  caused  partly  by  irradiation 
of  the  sun's  light,  and  partly  by 
the  irregularities  of  the  moon's 
surface.  An  occultation  may  occur 
at  the  summit  of  a  lunar  moun- 
tain, as  at  a,  Fig.  25,  or  in  the 
valley  near  it,  as  at  6.  The  re- 
corded times  of  these  two  observa- 
tions would  be  different,  of  course,  when  we  might  expect  them  to  be 
nearly  alike,  resulting  in  different  values  from  the  equations  of  con- 
dition for  the  semidiameter  or  for  the  constant  k.  And  it  follows 
that  the  constant  k  can,  therefore,  be  determined  more  accurately  for 
eclipses  than  for  occultations. 

It  would  be  difficult  to  measure  the  lunar  mountains,  because  it  is 
not  their  altitude  from  the  moon's  surface  that  is  needed,  but  their 
height  measured  on  a  secant  line  parallel  to  the  direction  of  the 
moon's  motions.  And,  moreover,  the  libration  of  the  moon  will 
bring  other  parts  of  the  disk  into  view  at  different  seasons. 

In  a  total  eclipse,  as  the  time  of  totality  approaches,  the  thin 
crescent  of  the  sun  breaks  up  into  a  number  of  dots  or  points  of 
light,  which  have  been  named  "Baily's  Beads,"  from  their  discoverer. 
They  are  DOW  known  to  be  caused  by  the  sun  shining  between  the 
lunar  mountains,  the  mountains  themselves  hiding  portions  of  the 
disk  of  the  sun.  It  is  seen  that  the  time  of  an  occultation  on  a 
lunar  mountain  will  be  when  the  beads  are  found  at  any  one  place, 
and  an  occultation  in  a  valley  will  be  the  time  when  the  beads  sever- 
ally disappear ;  so  that  if  the  time  of  continuance  of  these  beads  be 
observed  at  different  portions  of  the  moon's  disk,  the  height  of  the 
lunar  mountains  can  be  computed  in  the  direction  of  the  secant  line 
above  alluded  to.  From  a  number  of  observations  a  mean  value 
may  be  taken  as  the  moon's  semidiameter  in  occultations.  The 


182  THE   CONSTANT  k.  191 

dotted  curve  is  supposed  to  represent  the  sun's  disk  broken  up 
into  Baily's  beads. 

If  the  times  of  the  formation  and  disappearance  of  Baily's  beads 
have  been  observed,  and  the  times  predicted  accurately,  we  may  find 
the  effect  of  these  two  times  upon  k  in  the  following  manner. 

The  formula  for  the  tables  and  prediction  are  as  follows  : 

.  sin  H  —  k  sin  nf 

Eq.  (35)  sm/= — 

r'g 

(36)  c=z-^- 

sm/ 

k 
(38)  1=  ctan/=ztan/  — 


cos/ 
(267)  .L=J—  Ctan/ 


(268) 


(269)  r  =  db  _  m  cos        ~ 


in  which  &,  c,  £,  jL,  ^,  and  r  are  variables,  depending  upon  &  primarily 
and  upon  one  another.  In  the  first  of  these,  a  change  of  k  from  the 
value  0.272274  to  0.272509  affects  the  seventh  place  of  logarithms 
of  sin  /  by  only  nine  units  of  the  last  place  ;  /  is  a  small  angle  of 
about  17'  of  arc,  and  its  cosine  differing  but  little  from  unity  that  it 
may  be  so  taken  in  the  following  equations.  The  second  equation  is 
contained  in  the  next  following. 

Differentiating  the  last  four  successively  and  substituting  values 
from  previous  equations,  we  have  as  follows  :  Taking  here  the  lower 
sign  for  total  phase 

J1 

=  —  dk 


cos/ 
dL**dl**—dk 

,  msiu(M—N},T  LsiiKp,, 

cos  4>  d</>  =  --      -^—  -  -  dL  =  H  --  -  —  dk 
TJ  L 

,  .  tan  4>  7, 


, 
dr  = 


L 

cos  0  dL  —  L  sin  <J>  d</> 


n 


cos  (p  dk       sin  4'  tan  ^  die 

n  n 

cos2  0  -f  sin2 


dk 


n  cos  < 
Therefore,  dk  =  n  cos  <p  dr 


192  THEORY   OF   ECLIPSES.  182 

As  dr  may  be  observed  in  seconds  and  decimal,  whereas  n  is  given 
for  one  minute  in  the  formulae  for  prediction,  the  former  quantity 
must  be  divided  by  60.  Hence,  we  have 


(356) 


When  the  times  of  the  eclipse  have  been  correctly  computed 
beforehand,  this  formula  can  be  used  for  giving  the  difference 
between  the  two  values  of  k  for  eclipses  and  occultations,  dr  being 
the  interval  between  the  formation  of  Baily's  beads  and  their  disap- 
pearance ;  and  dk  the  corrections  for  the  eclipse  value  of  k,  which 
will  always  increase  the  value  for  occultations.  Cos  <f>  and  r  in 
an  eclipse  are  always  both  negative  for  beginning  and  both  positive 
for  ending,  so  that  if  taken  with  these  signs,  dk  will  always  be  a 
positive  quantity,  increasing  the  semidiameter  of  the  moon  in 
occultations. 

If  the  times  have  not  been  accurately  computed  for  the  place  of 
observation,  equations  of  condition  must  be  resorted  to  for  the  two 
times  of  formation  and  disappearance  of  Baily  beads. 

The  suggestion  is  made  that  as  the  lunar  mountains  are  of  various 
heights,  the  values  of  dky  resulting  from  numerous  observations,  will 
likewise  vary  between  wide  limits,  and  their  adjustment  must  rest 
with  the  judgment  of  the  computer,  for  these  discrepancies  are  not 
errors  to  be  reconciled.  Probably  a  mean  value  will  be  the  most 
generally  correct.  For  this  reason  also  occultations  can  never  be 
predicted  with  the  accuracy  of  an  eclipse. 

OUDEMAN'S  value  of  &,  0.27227  for  eclipses,*  is  derived  from  the 
least  value  of  the  moon's  semidiameter,  measured  in  the  valleys  of 
the  moon.  The  mean  value  of  the  semidiameter  is  1".25  greater, 
giving  the  constant  k  =  0.27264  for  occultations. 

J.  PETERS  f  has  more  recently,  from  a  large  number  of  occulta- 
tions, deduced  the  value  0.272518  for  occultations,  and  this  value  or 
values,  not  much  different,  have  recently  been  adopted  in  the  Eng- 
lish and  American  Nautical  Almanacs  for  occultations.  For  eclipses 
OUDEMAN'S  value,  0.27227  or  272274,  has  been  used  in  the  Ameri- 
can Nautical  Almanac  for  many  years,  except  the  years  1902-3-4. 
The  English  Nautical  Almanac  for  1905  has  adopted  very  nearly 
the  same  value,  by  taking  the  moon's  semidiameter  1".18  less  than 
for  occultations. 


*  A«tron.  Nach.,  li.,  pp.  25-6-30. 
t  Ibid.,  cxxxix.,  Nos.  3296-7. 


183  THE  CONSTANT  k.  193 

Equation  (356)  can  easily  be  shown  graphically.  In  Fig.  25  the 
height  of  the  lunar  mountains,  measured  in  the  direction  from  the 
centre  (7,  is  c  a  =  d  k,  the  increase  of  the  moon's  semidiameter.  a  e 
being  the  direction  of  the  moon's  motion,  Ca  e  =  <p  obtuse  for  the 
beginning  of  the  eclipse,  and  a  d,  measured  along  the  secant  line,  is 
the  effect  which  the  lunar  mountain  produces  upon  the  time  of  an 
occultation  ;  and  a  d  being  the  hypothenuse  of  a  right-angled  tri- 
angle, of  which  a  c  is  one  side,  we  have — 

dk 

ad  = 

cos^ 

and  the  time  of  describing  this  distance  is  found  by  dividing  it  by  n, 
the  space  passed  over  by  the  moon  in  one  minute,  giving  d  T.    Hence, 

dk 

dr  = 

n  cos  (p 

which  is  equation  (356). 

The  irradiation  of  light,  as  when  a  bright  surface  is  seen  upon  a 
dark  one,  is  also  known  to  increase  the  apparent  size  of  the  bright 
surface,  and  thus  to  cause  an  apparent  diminution  of  the  moon's 
diameter  in  solar  eclipses.  This  irradiation  is  the  explanation  given 
to  the  "  Black  Drop  "  in  Transits  of  Venus.* 

The  black  drop  can  be  seen  by  any  one  by  a  simple  experiment. 
When  looking  at  the  sky  when  bright,  or  the  white  shade  of  a  gas- 
lamp,  hold  the  thumb  and  fingers  before  one  eye,  and  bring  them 
slowly  together;  just  as  they  are  about  to  touch,  a  little  swelling 
will  appear  on  each,  that  move  toward  one  another.  It  can  some- 
times be  better  seen  if  the  finger  is  held  close  to  the  eye. 

183.  There  is  one  point  in  the  theory  of  eclipses  and  occultations 
that  is  rather  obscure  in  the  formulae,  and  therefore  perhaps  not  gen- 
erally recognized,  which  is,  that  the  earth  in  proportion  to  the  moon 
is  taken  smaller  in  eclipses  than  in  occultations.  This  led  me  to 
think  that  it  had  some  effect  upon  the  constant  k,  but  there  is  a  com- 
pensation in  the  formulae,  so  that  k  is  not  affected  by  it.  As  this 
investigation  reveals  some  points  not  otherwise  explained,  and  also 
shows  clearly  the  differences  in  the  formulae  when  used  for  eclipses 
and  occultations,  it  may  not  be  amiss  to  give  it  a  place  in  this  section. 

If  we  wish  to  ascertain  the  difference  of  apparent  declination  be- 
tween the  sun  and  moon,  we  apply  their  parallaxes,  thus : 
(5  _  TT)  —  (d'  —  TT')  =  d  —  (TT  —  TT')  —  d'. 

*  This  is  described  in  Professor  NEWCOMB'S  Popular  Astronomy,  Ed.  1878,  p.  179 
et  seq. 

13 


194  THEORY  OF   ECLIPSES.  183 

In  the  second  member  we  have  applied  the  two  parallaxes  to  the 
moon,  while  the  sun  is  taken  in  its  true  position,  being  given  by  its 
centre.  This  difference  of  parallaxes  has  been  styled  by  Professor 
LOOMIS  the  relative  parallax. 

In  eclipses  the  same  thing  is  done,  which  is  illustrated  in  Fig.  26. 
Let  E  be  the  centre  of  the  earth,  8  that  of  the  sun,  and  M  the  semi- 

FIG.  26. 

iM 


c 


_"— | 

"71 


S  r  O  E 

diameter  of  the  moon.  The  shadow  will  first  touch  the  earth  at  6, 
when  the  moon's  limb  is  at  ra  on  the  line  efy  drawn  tangent  to  the 
disks  of  the  sun  and  moon.  But  if  we  attribute  the  sun's  parallax 
to  the  moon,  the  sun  becomes  analytically  a  point  at  S,  and  the  eclipse 
will  commence  when  the  moon  thus  transformed  reaches  the  point  n 
on  the  line  e  S,  drawn  to  the  centre  of  the  sun.  We  now  require  to 
know  the  length  n  0,  which  is  in  proportion  to  e  E  as  their  distance 

from  S. 

nO:eE::rf  —  r:r'. 

Taking  for  e  E  the  parallax  TT,  we  have — 


r' 
But  by  (23)  r  = >          or  nearly  r  =  - ;  and  rf  =  — 

SID  7T  7T  7Tr 

Whence     n  0  =  n  —  TT'. 

This  is  the  radius  of  the  cone  of  solar  parallax  at  the  distance  of 
the  moon  from  the  earth,  and  it  represents  the  earth's  radius  in 
BESSEL'S  Theory  of  Eclipses.  Professor  LOOMIS,  in  his  Practical 
Astronomy,*  seems  to  have  recognized  the  facts  here  presented,  but 
not  clearly.  In  treating  of  Eclipses  of  the  Sun,  Chapter  XL,  Art. 
249,  he  says:  "The  relative  parallax  is  54'  19".l,  or  3259".!,  which 
represents  the  apparent  semidiameter  of  the  earth's  disk,  if  seen  at  the 
distance  of  the  moon  from  the  earth."  This  is  merely  the  definition 
of  the  moon's  parallax.  The  relative  parallax  cannot  be  seen,  since 

*  An  Introduction  to  Practical  Astronomy,  -by  ELIAS  LOOMIS,  LL.D.  Harper  and 
Brother,  New  York,  7th  Edition,  1873. 


183  THE   CONSTANT  k.  195 

it  is  only  the  radius  of  the  cone  of  solar  parallax  at  the  distance  of 
the  moon  from  the  earth. 

We  will  now  show  this  quantity  in  CHAUVENET'S  formulae.  It 
also  occurs  in  Section  XIX.  on  the  Method  by  Semidiameters.  In 
Art.  113,  for  central  eclipse  at  noon,  we  derived  the  following 

formula  : 

I  3  —  d' 


which  is  rigorous,  excepting  only  that  the  arc  is  written  for  the 
sine.  And  this  is  still  further  reduced  in  Art.  159,  on  the  projec- 
tion by  the  Method  of  Semidiameters,  by  showing  that  n  (1  —  b)  = 

TT  —  ~',  or 

y-^4  (357) 


This  formula  being  computed  for  the  central  eclipse  at  noon,  y  lies 
in  the  axis  of  coordinates,  and  can  never  give  real  values  on  the  earth 
if  it  exceeds  unity,  therefore,  n  —  TT'  must  represent  the  earth,  the 
unit  of  measure.  If  the  formulae  for  x  and  z  be  similarly  reduced, 

cos  8  sec  d'  (a  —  d) 


x  = 

7T 

cos  (d  — 


(358) 


All  of  which  are  fractions  having  the  same  denominator,  which  rep- 
resents the  earth's  radius  in  this  theory.  At  the  instant  of  conjunc- 
tion, however,  a  —  a  =  0  and  x  =  0. 

The  effect  of  diminishing  the  denominator  of  a  fraction  is  to  in- 
crease its  value,  and  this  is  the  effect  produced  numerically  upon  x  y 
and  z  in  eclipses,  which  can  readily  be  shown,  thus  :  taking  the  quan- 
tity a  —  a,  upon  which  x  mainly  depends,  we  know  that  it  is  the 
sum  of  (a  —  a'),  and  the  small  term  of  equations  (17  and  18).  What 
quantity  subtracted  from  IT  will  be  the  equivalent  of  this  small  term  ? 

(«  —  a)  :  small  term  : :  K  :  x. 

Taking  («  —  a)  from  the  Eclipse  Tables,  Art.  37,  n  from  the  Data, 
Art.  17,  and  the  logarithm  of  the  small  term  from  the  example,  Art. 
36,  all  for  12A,  we  have— 

«  —  a  _  small  term    1".48  58".52  __  15".57  log  (1.19139) 
*  x  61'  22.600  8.75~~ 

The  sun's  parallax  for  this  date  is —  8.74 

That  this  result  just  equals  the  solar  parallax  js  what  we  wished 
to  prove.  We  see  now  that  in  order  to  take  for  the  earth  a  smaller 


196  THEORY   OF   ECLIPSES.  183 

value,  the  coordinates  are  increased,  which  has  the  same  effect.  In 
occupations  these  small  terms  are  equal  to  zero,  making  the  coordi- 
nates smaller  than  in  eclipses,  which  has  the  same  effect  as  making 
the  earth  larger.  The  moon's  parallax  remains  the  same  in  both. 
We  thus  learn  the  meaning  of  these  small  terms,  and  thereby  arrive 
at  a  better  understanding  of  both  eclipses  and  occultations. 

In  Section  XXVI.  on  Occultations,  the  formulae  for  the  coordi- 
nates are  (Equation  436) : 

cos  d  sin  (a  —  a') 

x  = 

sin  TT 

sin  (d  —  8')  cos2  Ka  —  «0  +  sin  (d  -f-  3')  sin2  %  (a  —  a') 

sin  TT 

If  these  are  reduced  in  the  same  manner  as  those  for  eclipses  (357 
and  358),  we  have — 

COS  d  (a  —  a') 

x  ^= 


z  is  infinity  in  occultations. 

which  are  similar  to  the  equations  for  eclipses  with  the  change  in  the 
denominators  as  above  noted. 


SECTION    XXII. 

CORRECTIONS  FOE  REFRACTION  AND  ALTITUDE. 

184.  Correction  for  Refraction. — If  the  sun  is  low  in  the  horizon 
during  an  eclipse,  the  refraction  of  the  atmosphere  will  much  affect 
the  times.  Professor  CHATJVENET  has  discussed  this  in  the  follow- 
ing manner  in  his  chapter  on  Eclipses,  Art.  327,  p.  515  :  A  ray  of 
light  before  reaching  the  earth  is  refracted  toward  the  normal  to  the 
surface,  and  since  the  atmosphere  increases  in  density  as  the  ray 
approaches  the  earth,  its  path  will  be  a  curve  concave  toward  the 
earth.  This  ray,  we  will  suppose,  reaches  an  observer  at  some 
point  (A)  on  the  earth's  surface,  and  he  sees  the  sun  apparently 
higher  in  the  heavens  than  it  really  is.  If  there  were  no  refraction, 
the  ray  of  light,  if  continued  in  a  straight  line,  would  reach  the  ver- 
tical line  above  A  at  some  point  (J9) ;  at  which  point,  if  there  were 
no  refraction,  an  observer  would  see  a  true  contact  at  the  same 
instant  as  the  observer  at  B  sees  the  contact  of  the  refracted  ray. 


184  CORRECTIONS  FOR  REFRACTION  AND  ALTITUDE.  197 


CHAUVENET'S  method  of  taking  account  of  the  refraction  is  to 
substitute  the  point  B  for  A  in  the  formulae  for  the  position  of  the 
observer.  Let  s  be  the  height  of  B  above  A  and  p  the  radius  of  the 
earth,  as  in  all  the  preceding  formulae.  It  is  then  only  necessary  to 
substitute  p  -j~  s  instead  of  p  in  the  formulae  for  £,  37,  and  £,  such  as 
equations  (131),  (132),  Art.  81  ;  (261),  Art.  150,  etc. ;  or  else  more 

simply  write  p  1 1  + -I  for  p.  "Therefore,"  says  Professor  CHAUVE- 
NET in  Art.  327,  "when  we  have  computed  the  values  of  log  f,  log  y, 
and  log  f  by  those  equations  [(483)  of  his  work]  in  their  present 
form,  we  shall  merely  have  to  correct  them  by  adding  to  each  the 

value  of  k 

This  logarithm  he  has  computed  by  the  following  equation,  of 
which  he  gives  the  derivation  : 


1  + 


sin  Z 


(359) 


in  which  Z  is  the  true  zenith  distance  of  the  sun  or  ray  of  light, 
and  Zf  the  apparent  zenith  distance,  and  p^  the  index  of  refraction 
of  the  air  at  the  observer. 

The  following  table  is  given  by  CHAUVENET,  computed  for  the 
values  of  the  mean  refraction  tables  given  in  his  work  ;  that  is,  ft 
and  Y  of  the  tables  each  =  1  and  a  mean  value  of  fo(=  1.0002800). 
Log  £  is  the  argument  and  log  £,  log  ^,  and  log  £  are  each  to  be  cor- 
rected by  the  same  amount,  taken  from  the  following  table,  which  he 
has  deduced  from  BESSEL  : 


CORRECTION  FOR  REFRACTION  FOR  LOG  ?,  LOG  3 
(CHAUVENET'S  Astron.,  i.,  Art.  327). 


AND  LOG 


log  <    Correction  for  logs 
of  f  ,  -n,  & 

1     ^    Correction  for  logs 
of  £,  •»?,  £ 

j     ^    Correction  for  logs 
of  f  ,  TJ,  £. 

0.0 

0.0000000 

9.0 

0.0000119 

8.0 

0.0000788 

9.9 

0001 

8.9 

0167 

7.9 

0835 

9.8 

0002 

8.8 

0225 

7.8 

0875 

9.7 

0005 

8.7 

0292 

7.7 

0909 

9.6 

0008 

8.6 

0367 

7.6 

0937 

9.5 

0.0000014 

8.5 

0.0000446 

7.4 

0.0000978 

9.4 

0023 

8.4 

0525 

7.2 

1006 

9.3 

0035 

8.3 

0602 

7.0 

1023 

9.2 

0054 

8.2 

P672 

6.5 

1044 

9.1 

0081 

8.1 

0734 

6.0 

1051 

9.0 

0.0000119 

8.0 

0.0000788 

00 

0.0001054 

198  THEORY   OF   ECLIPSES.  185 

185.  Correction,  for  the  Altitude  of  the  Observer  above  the  Sea-level. 
— If  sr  is  the  altitude  of  the  observer  above  the  sea,  it  is  only  neces- 
sary to  substitute  p  +  s'  instead  of  p  in  the  general  formulae  of  an 
eclipse,  which  is  simply  done  by  adding  to  log  c,  log  y,  and  log  f  the 

value  of  log!  1 -f —V  From  the  theory  of  logarithms  we  have 
generally 

log  (l  +  ^)  =  log  P  =  M (^  +  j  gj  +,  etc.)  (360) 

in  which  M  is  the  modulus  of  the  system.  The  second  term  is  small 
compared  with  log  /?,  so  that  all  the  terms  within  the  parentheses 
except  the  first  may  be  omitted ;  and  assuming  for  log  p  a  mean 
value  for  latitude  =  45°,  we  have  for  sf ,  when  expressed  in  English 
feet, 

Correction  for  log  £,  log  77,  and  log  C  =  0.000  000  020  79  s'    (361) 

If  s'  is  expressed  in  meters,  we  have  for  the  constant,  instead  of  the 
above, 

0.000  000  064  s'  (362) 

For  example,  if  the  observer  is  1000  feet  above  the  sea,  the  cor- 
rection is  0.0000208,  to  be  added  to  log  £,  log  ^,  and  log  £. 


SECTION    XXIII. 

SAFFOED'S  TRANSFORMATION  OF  ECLIPSE  FORMULA. 

186.  THESE  formulae  were  used  in  the  old  Almanacs  down  to  the 
year  1881  inclusive,  but  for  the  subsequent  years  BESSEL'S  notation 
has  been  restored.  In  BESSEL'S  notation  the  quantities  given  in  the 
eclipse  tables  of  the  Nautical  Almanac  have  an  astronomical  signi- 
fication, but  by  this  transformation  the  quantities  on  the  other  hand 
have  not.  BESSEL'S  elegant  formulae,  instead  of  being  simplified,  are 
only  obscured.  The  transformation  is  by  Professor  TRUMAN  HENRY 
SAFFORD,*  who  was  engaged  upon  the  Nautical  Almanac  during 
some  of  its  earlier  years. 

*  These  are  given  by  CHAUVENET,  but  had  previously  been  printed  by  Professor 
BENJAMIN  PEIRCE  in  his  Spherical  Astronomy. 


186     SAFFORD'S   TRANSFORMATION   OF   FORMULAE.     199 

In  the  fundamental  equation  of  eclipse 

(x  -  £)2  =  (I  -  ^  ^n  /)2  -  (y  -  i?)2  (363) 

Assume  a2  =  62  —  c2  (364) 

Then  a  =  x  —  $  (365) 

c=(J-C  tan  /)-(y-,)  (367) 

We  have  for  the  coordinates  of  the  given  places 

£  =  /?  cos  ^'  sin  i9  "| 

TJ  =  ^o  sin  ?>'  cos  d  —  ,0  cos  ?>'  sin  d  cos  *  V  (368) 

C  =  /?  sin  ?>'  sin  d  -I-  p  cos  ^'  cos  d  cos  d  J 

which  are  the  principal  equations  from  which  the  various  equations 
for  the  coordinates  have  been  derived  in  the  foregoing  sections,  as, 
for  example,  equation  (361),  Art.  150,  in  prediction.  Substituting 
these  values  in  the  two  previous  equations,  they  become 

b  =  I  +  y  —  p  sin  <pr  (cos  d  -f  sin  d  tan  /) 

+  p  cos  <pf  (sin  d  —  cos  d  tan  /)  cos  # 
c  =  I  —  y  +  p  sin  <p'  (cos  d  —  sin  d  tan/)  ' 

—  p  cos  tpr  (sin  d  -f  cos  d  tan  /)  cos 
Place  -4  =  x 


C=-l  +  y 

E  =  cos  d  +  sin  c?  tan/=  cos  (d  — /)  sec/ 
F  =  cos  c?  —  sin  d  tan  /  =  cos  (d  -f-  /)  sec  / 
6r  =  sin  d  —  cos  d  tan  /  =  sin  (d  —  /)  sec  / 
H  —  sin  d  -f  cos  d  tan /  =  sin  (d  +  /)  sec/ 

We  also  have,  as  in  the  previous  method,  p.  (//x)  and 
,      J/£,  sin 


(370) 


3600 


106  for  the  meridian  of  Washington.          (371) 


All  of  these  eight  quantities  last  given  are  independent  of  any 
place  on  the  earth,  and  constitute  the  eclipse  tables  of  xthis  method. 
The  tables  also  give  A1  ',  B',  C'.  The  changes  of  A,  B,  C,  in  one 
second,  and  also  5,  (7,  E,  F,  G,  H  for  shadow  ;  or  as  E,  F,  G,  H 
differ  from  those  quantities  for  penumbra  by  a  small  constant,  this 
constant  is  given  instead  of  the  quantities  themselves.  It  is  to  be 
noted  also  that  in  the  Nautical  Almanac  the  quantities  Ar,  Bf,  0'9 
and  fj.f  the  changes  of  p  are  given  in  units  of  the  sixth  decimal,  so 
that  to  form  them  these  quantities  in  the  formula  of  this  section,  as 
well  as  //  in  equation  (376)  and  r  in  equation  (381),  must  be  multi- 
plied by  106. 


200  THEORY   OF   ECLIPSES.  187 

187.  Prediction  by  this  Method. — Substituting  the  values  of  (370) 
in  (369),  we  have 

a  =  x  —  £  =  A  —  p  cos  <p*  sin  $  "j 

b  =  B  —  Ep  sin  <f>'  +  Gp  cos  y'  cos  #       >  (372) 

c  =  —  (7  +  -F/o  sin  <p*  —  Hp  cos  ?'  cos  #  J 


And  the  fundamental  equation  becomes 

a  =  Vfc  (373) 

For  any  assumed  time  TQ9  take  from  the  eclipse  tables  the  proper 
quantities  and  compute  a,  6,  and  c  by  equation  (372),  and  if  a  differs 
from  l/6c,  the  assumed  time  requires  a  correction,  to  be  found  as 
follows  : 
Place  m  =  Vfo  (374) 

a',  &',  c'  =  the  changes  of  a,  b,  and  c  in  one  second. 

r  =  the  required  correction  for  the  assumed  time  T0. 


Whence          r  =    m  "~  a  (375) 

a'  —  mr 

To  find  a/,  differentiate  equation  (372),  remembering  that  as  co  = 
fa  —  $,  dfa  =  d&,  and  distinguishing  the  derivatives  by  accents, 

a'  =  A'  —  n'p  cos  <p'  cos  *  x  106  (376) 

And  to  find  m', 

JB'=2/'=C'' 
bf  =  Br  —  fj.fGp  cos  ?'  sin  * 
c'  =  —  Cf  +  fj.'Hp  cos  <pf  sin  i? 

Since  /  is  small  in  these  expressions,  we  may  place  G  =  H.    Hence, 

V  =  -  cf  =  Br  —  IJL'  Gp  cos  ?'  sin  *  (377) 

And  from  (374) 


(378) 
c 

Assume  tan  \  Q  =  \/-  =  —  =  2.  (379) 

^6        m        6 

m'  =  —  b'  cot  §  (380) 


187     SAFFORD'S   TRANSFORMATION   OF   FORMULA.     201 
Then  r  is  found  from  the  following  : 


In  applying  these  formulae  for  prediction 
Compute  a,  b,  c,  by  (372). 

m  by  (374)  (which  usually  has  the  same  sign  as  a). 
a'  by  ((376)  and  bf  by  (377). 
tan  i  Q  by  (379). 
and  finally  r  by  381. 

And  the  correct  time  is  in  Washington  mean  time. 

T=T0+T  (382) 

Or  in  local  mean  time,  to  being  the  longitude  from  Washington. 

T=  TO+T  —  «>  (383) 

With  this  time  a  second  approximation  can  be  made. 
The  angle  Q  here  is  the  same  quantity  as  that  used  throughout 
the  preceding  pages,  for  we  have  from  (380)  at  the  instant  of  contact. 

_  V         2m 

tan  Q  =  --  7=7— 

mr       b  —  c 

And  from  (365-6-7) 

tan  Q  =  ^=1  (384) 

y  —  -n 

Q  has  the  same  sign  as  a  for  the  penumbra  or  umbra  of  an  annular 
eclipse,  but  has  a  contrary  sign  for  the  shadow  of  a  total  eclipse.    As 
before,  it  is  measured  from  the  north  point  toward  the  east. 
The  angle  from  the  vertex  is  given  by  the  following  equations  : 
p  sin  P  =  sin  <f> 
p  cos  P  —  cos  <p  cos  $ 
c  sio  C  =  cos  P  tan  & 
c  cos  C  =  sin  (P  —  3f) 

V=  Q-C  (385) 

d'  here  is  the  sun's  declination. 

The  magnitude  of  the  eclipse  is  found  as  follows,  being  given  in 
the  antiquated  digits. 

Place  20  =  The  difference  of  the  two  values  of  Q  for  beginning 
and  ending,  having  regard  to  signs. 

s        semidiameter  of  the  moon 

n  ==  —  =  -  —  --- 

sr       semidiameter  of  the  sun 
M=  f  12  (1  +  n)  sin2  \  0  if  0  is  acute    ) 

"  1  12  (1  +  n)  cos2  \6  if  e  is  obtuse  j 

By  omitting  the  constant  12,  the  magnitude  as  usually  now  given 
will  be  expressed  as  a  fraction  of  the  sun's  diameter. 


PAET  II. 
SECTION    XXIV. 

LUNAK  ECLIPSES. 

188.  Introduction. — Criterion. — This  subject  comprises  Arts.  338 
and  339  in  Professor  CHAUVENET'S  chapter  on  Eclipses.  There  is 
much  similarity  between  the  formulae  for  solars  and  lunars,  so  that 
reference  to  the  former  will  often  explain  a  kindred  point  in  this 
section.  Lunar  eclipses  cannot  be  observed  with  the  same  degree  of 
accuracy  as  solar — the  approach  of  the  eclipse  not  being  so  well 
marked — for  which  reason  they  are  considered  of  much  less  impor- 
tance, and  the  formulae  consequently  are  not  so  rigorously  exact  as 
those  in  the  preceding  sections.  The  earth's  atmosphere  refracts 
some  of  the  sun's  rays  on  the  boundaries  of  the  cone  of  shadow, 
making  it  larger  than  the  true  diameter  of  the  earth  would  cause. 
This  is  estimated  as  one-sixtieth  in  LOOMIS'  Astronomy,  and  one-fif- 
tieth in  CHAUVENET'S  chapter.  As  the  true  amount  of  this  refrac- 
tion is  not  known,  the  error  cannot  be  determined.  The  spheroidal 
form  of  the  earth  is  another  source  of  error,  which  would  necessitate 
a  second  approximation  ;  but  as  this  is  small,  the  earth  is  taken  as  a 
sphere  of  the  radius  of  latitude  45°,  which  is  too  small  at  the  equa- 
tor and  too  large  at  the  poles.  And  as  the  refraction  in  the  first  case 
would  cause  a  gradual  diminution  of  the  sun's  rays,  this  is  undoubt- 
edly the  reason  why  a  lunar  eclipse  cannot  be  observed  as  accurately 
as  a  solar  eclipse — the  moon  being  gradually  obscured. 

The  criterion  for  lunar  eclipses  is  similar  to  that  of  solar  (Art.  8), 
except  that  here  we  have 

ft  COS  F  <  ft  (TT  -  «'  +  «')  +  3  (387) 

or  with  a  mean  value  of  I'. 

0  <  Ki  (*-«'  +  *')  +  «]  X  1.00472 
The  small  term  varies  from  15".3  to  18".0,  or  with  a  mean  value. 

0  <  tt  (*-«'+*')  +  «  +  16".6  (388) 

202 


188  LUNAR   ECLIPSES.  203 

With  the  values  adopted  in  Art.  10,  we  have 

/?  >  63'  51"  Eclipse  impossible.  )  (389") 

/?  <  53'  29"  Eclipse  certain.       3 

Between  these  limits  eclipse  doubtful. 

These  values  differ  somewhat  from  CHAUVENET'S,  but  it  is  be- 
lieved that  they  are  more  correct. 

As  in  solar  eclipses,  the  best  way  to  ascertain  what  eclipses  there 
will  be  in  a  certain  year  is  by  means  of  the  Saros  (Art.  11)  ;  but  this 
method  will  not  show  a  new  series  of  eclipses  entering,  for  which  the 
above  criterion  must  be  employed,  or  else  in  close  cases  proceed  as  if 
there  were  an  eclipse  and  compute  it ;  and  if  sin  <p  is  less  than  unity 
there  will  be  an  eclipse. 

The  motion  of  the  series  of  successive  eclipses  is  governed  by  the 
moon's  node  as  in  solars.  At  the  moon's  ascending  node  the  series 
is  moving  south,  and  at  the  descending  node  it  is  moving  north,  the 
quantity  d  +  dr  at  the  time  of  opposition  shows  the  amount  of  mo- 
tion of  successive  eclipses.  If  the  eclipse  is  known  to  be  north  of 
the  equator  and  moving  north,  for  example,  the  durations  of  each 
successive  eclipse  are  decreasing.  If  the  series  is  decreasing,  as 
shown  by  d  +  df,  the  durations  and  magnitudes  will  generally  also 
decrease ;  but  these  latter  are  no  standard  for  the  general  amount  of 
decrease  as  the  quantity  d  +  df  always  is.  A  curious  case  occurred 
in  the  year  1898,  the  partial  lunar  of  January  7.  The  series  was 
moving  north,  and  the  successive  eclipses  decreasing  at  the  north 
pole.  Compared  with  previous  eclipses  of  the  series  1879,  Decem- 
ber 27, 1  found  that  the  duration  of  the  shadow  had  increased.  Both 
eclipses  of  this  year  exhibited  the  same  peculiarity,  which  is  shown 
as  follows : 

Magnitude. 
Totality. 

.  .  .  0.202 
...  0.185 
...  0.164 
...  0.157 

1844,  Nov.  24,      6»  15-.2  _  2  -  3*  49W.8  1A  33ro.O  1.435 

1862,  Dec.  5,        6   12  .5   ,   2  4  3    49  .7  1    32  .0  1.415 

1880,  Dec.  15, 16,  6   14  .9  ^  2  9  3    48  .9  1    29.9  1.388 

1898,  Dec.  27,       6   17  .8  3   48  .9  1    29  .2  1.384 


Durations. 

Penumbra. 

Shadow. 

1843,  Dec.  6,         5*  8m.2       g  2 

1A  46W.2 

1861,  Dec.  16,       5    5  .0        ' 

1    42  .2 

1879,  Dec.  27,        5    4  .7   ,   7'« 

1    35  .9 

H-  i.o 

1898,  Jan.  7,         5  12  .5  ^ 

1    35  .5 

204  THEORY   OF   ECLIPSES.  188 

The  cause  of  this  eccentricity  seems  to  be  that  while  the  point 
given  by  (d  -\-  8')  in  the  axis  of  Y  properly  moved  north,  the  inclina- 
tion of  the  path  also  greatly  increased,  so  that  the  middle  point  was 
brought  nearer  to  the  centre  of  the  penumbral  shadow.  If  the 
inclination  had  increased  a  little  more,  the  duration  of  the  shadow 
of  the  December  eclipse  would  also  have  increased ;  and  it  seems 
possible  that  the  next  eclipse  of  the  series,  1917,  Jan.  8,  may  have 
an  increased  magnitude.  A  change  of  constants  used  would  make 
some  differences  in  the  times,  but  would  not  likely  make  the  differ- 
ence of  5m.3  shown  above.  Theoretically,  this  cause  alone  would 
have  increased  both  durations,  but  doubtless  the  other  changes  in  the 
elements  partially  counteracted  its  effects ;  for  example,  the  sun's 
semidiameter  acts  upon  the  shadow  and  penumbra  with  contrary 
signs.  The  circumstance  was  not  investigated  at  the  time.  The 
total  eclipse  of  Dec.  27,  of  the  same  year,  1898,  exhibits  a  similar 
peculiarity. 

189.  General  Formulae. — The  notation  here  is  generally  the  same 
as  in  solar  eclipses,  but  instead  of  the  sun  we  have  for  the  centre  of 
the  earth's  shadow, 

a'  =  Sun's  R.  A.  +  12*  (retained  in  time). 
d  =  Declination  of  the  shadow  =  —  sun's  decimation. 
L  =  Distance  between  centres  of  moon  and  shadow. 

In  the  spherical  triangle  formed  by  the  centres  and  pole 

L  sin  Q  =  cos  d  sin  (a  —  a')  ") 

L  cos  Q  =  cos  3f  sin  d  -\-  sin  3f  cos  d  cos  (a  —  a')  j 

As  the  earth  is  not  a  sphere,  a  mean  value  of  its  radius  at  45°  is 
substituted  for  the  variable  radius  in  computing  the  moon's  parallax, 
so  that  instead  of  TT  in  the  formulae,  we  have 

^  =  [9.99928]*  (391) 

And  the  radii  of  the  shadow  and  penumbra  increased  by  -^  are 

Radius  of  shadow       =  §£  (TT,  —  sf  +  r')  }  (392) 

Radius  of  penumbra  =  -f-J-  (^  -|-  sf  +  w')  J 

Therefore,  we  have  for  L 

First  and  last  contacts  with  penumbra,  L  =  |-J-  (TTJ  -f-  sr  +  TT')  -f  s  ^ 

shadow,     L  =  f  £  (*,  —  s'  +  «')  +  s\  (393) 
Second  and  third      "  "  L  =  ft  fa  —  sf  +  «')  —  s  ) 


189  LUNAR   ECLIPSES.  205 

The  second  and  third  contacts  with  penumbra  are  not  required. 
They  can  be  found  by  changing  the  sign  of  s. 

These  formulae  need  not  be  so  rigorously  exact,  as  above  remarked  ; 
we  may  write  the  arc  for  the  sine  in  (390),  which  gives 

L  sin  Q  =  (a  —  «')  cos  d  ~\ 


sin  1"  ) 

_  sin  2<?  sin2  £  (a  —  g/) 

sin  V 
=  [6.4357]  sin  2<?  (a  —  a')2  (395) 

Also  X  =  15  (a  —  a')  COS  d  "| 

y  =  d  +  d'  -  e  V  (396) 

x1  y'  =  The  hourly  changes  of  x  and  y  ) 
And  (394)  becomes 

L  sin  Q  =  x  +  xr 
L  cos  Q  =  ii  4-  vr 


To  solve  the  above  by  finding  r,  which  is  analogous  to  that  quantity 
in  solar  eclipses,  place 


AT  /  I  (3") 

HCOSN  =  y'  ) 

„,  m  sin  (If—  JV)  //1Ar., 

Then  sin  0  = «  (400) 

.L 

.L  cos  <l>      wi  cos  (Jf —  JV) 
r  = ^ (401) 

T=T0-fr  (402) 

Local  time  for  any  place,  T=  T0  +  r  —  a>  (403) 

As  in  solar  eclipses,  cos  <p  is  to  be  taken  with  the  negative  sign  for 
beginning,  giving  an  obtuse  angle ;  and  with  the  positive  sign  for 
ending,  giving  an  acute  angle. 

The  time  of  greatest  obscuration,  which  is  usually  also  considered 
the  middle  of  the  eclipse,  is 


Ti=Ta_- 
n 

The  angle  of  position  of  the  point  of  contact  on  the  moon's  limb, 
measured  from  the  north  point  toward  the  east, 

(405) 


206  THEORY   OF   ECLIPSES.  189 

The  least  distance  of  the  centres 

A  =  ±  m  sin  (M  —  N)  (406) 

the  double  sign,  implying  only  that  J  shall  be  positive.     And  the 
magnitude  of  the  eclipse 

M= 


in  which  the  value  of  L  for  total  shadow  is  to  be  used,  and  J  inter- 
polated for  the  middle  of  the  eclipse.  For  a  partial  eclipse,  M  is 
less  than  unity,  but  for  a  total  eclipse  it  is  greater  than  unity,  and 
shows  how  far  the  moon  is  immersed  in  the  earth's  shadow. 

Moon  in  the  Zenith.  —  The  formulae  for  this  problem  are  not  given 
by  Professor  CHAUVENET,  nor  do  I  remember  seeing  them  in  any 
astronomy.  They  are,  however,  easily  derived  as  follows  :  The 
Greenwich  mean  time  at  any  instant  gives  the  longitude  of  the 
meridian  of  the  mean  sun,  T.  By  adding  the  equation  of  time,  E 
(reducing  from  mean  to  apparent  time),  we  have  the  longitude  of  - 
the  meridian  of  the  true  sun,  T  -j-  E;  the  meridian  of  the  centre  of 
the  shadow  being  12  hours  greater,  or  T  -\-  E-\-  12h.  As  the  moon 
in  the  heavens  moves  toward  the  east,  while  longitudes  are  measured 
from  the  east,  the  moon's  position  at  the  beginning  of  an  eclipse  will 
numerically  increase  the  above  quantity,  and  decrease  it  for  ending. 
The  distance  of  the  moon  from  the  centre  of  the  shadow  is  (a  —  «'), 
which  is  negative  for  beginning  and  positive  for  ending  ;  hence,  it 
must  be  algebraically  subtracted  from  the  above.  Hence,  we  have 
the  moon's  longitude  in  arc  at  the  time,  T. 

<*=  15  [T+E'+  12*  -(«-«')]  (408) 

(a  —  a')  must  be  interpolated  for  the  time  T  ';  and  the  factor  15  is 
merely  the  reduction  from  time  to  arc. 

The  latitude  is  simply  the  moon's  declination  at  the  times  T. 

<P=3  (409) 

These  are  the  positions  of  the  places  on  the  earth  which  have  the 
moon  in  the  zenith,  for  the  first  and  last  contacts  with  the  moon's 
limb.  They  are  not  used  for  the  other  times  ;  and  they  give  the 
zenith  of  the  earth's  hemisphere  from  which  the  contacts  can  be  seen. 

> 

190.  Data,  Elements,  Example.  —  The  data  for  a  lunar  eclipse  are, 
with  one  exception,  the  same  as  those  for  solars  (Section  III.)  ;  the 
right  ascension,  declination,  semidiameter,  and  parallax  of  the  sun 
and  moon,  interpolated  from  the  Nautical  Almanac,  for  at  least  three 


190 


LUNAR   ECLIPSES. 


207 


hours  before  and  after  the  time  of  opposition  in  right  ascension. 
But  instead  of  the  sidereal  time,  we  here  require  the  equation  of  time 
from  page  II.  for  the  month  from  the  Nautical  Almanac,  and  for  the 
preceding  and  succeeding  noons.  To  find  the  time  of  opposition 
approximately,  the  most  convenient  method  is  by  the  Saros,  Art.  11, 
in  Solar  Eclipses ;  but  a  new  series  of  eclipses  may  enter,  so  that 
search  should  be  made  at  suspected  places  in  the  Ephemeris,  by 
means  of  the  Criterion. 

The  right  ascension  should  be  interpolated  to  the  decimals  of  a 
second,  and  it  is  more  convenient  to  retain  them  in  time,  though 
CHAUVENET  reduces  them  to  arc — one  decimal  of  a  second  of  arc  is 
sufficient  throughout  this  work. 

There  being  no  Lunar  Eclipse  in  the  year  1904,  from  which  the 
example  for  the  solar  eclipse  in  the  foregoing  pages  was  taken,  an 
example  here  is  selected  from  the  year  1902.  The  data  above  men- 
tioned are  not  here  given  separately,  but  placed  in  the  example 
below,  where  they  can  be  more  conveniently  made  use  of.  Differences 
also  are  here  omitted  for  want  of  space. 

TOTAL  LUNAR  ECLIPSE,  1902,  APRIL  22. 
EXAMPLE,  PRELIMINARY  COMPUTATION. 


G.  M.  N 

.      0  App.  R. 

A. 

D  App. 

R.  A. 

(a  -a') 

log  (a  —  a'). 

cos  S. 

logs. 

3* 

1*  57™  3P.96 

13ft50m7*.32 

_7m  24*. 

64 

—2.64801 

+9.99074 

—3.81484 

4 

41 

.30 

52 

7.31 

5    33. 

99 

.52374 

050 

.69036 

5 

50 

.65 

54 

7.36 

3   43. 

29 

.34887 

031 

.51527 

6 

59 

.99 

56 

7.47 

1    53. 

52 

2.05123 

010 

.21743 

TO  7 

58     9 

.37 

58 

7.64 

—0      1. 

69 

—0.22789 

.98988 

—1.39386 

8 

18 

.68 

14 

0 

7.87 

+1    49. 

19 

+2.03819 

967 

+3.20395 

9 

28 

.03 

2 

8.16 

3   40. 

13 

.34268 

944 

.50821 

10 

1    58  37 

.37 

14 

4 

8.52 

+5   31. 

15 

2.52002 

+9.98922 

+3.68533 

X. 

x'. 

7T. 

(ir1+7r/).  (s—  Const.)  (s'—  Const.) 

3A—  6528.8    ,  1fi0«n    +1626.0 

54  43.59 

3287.0 

894.8 

954".33 

4 

4901.9            ^nrCs,  '  A 

5.7 

42.77 

6.2 

4.5 

.32 

5 

3275^5 

1OZO.4: 

5.3 

41.95 

JT> 

+8.75 

5.4 

4.3 

.31 

6 

1649,8 

1C9K  A 

5.0 

41.14 

cor 

—5.37 

4.6 

4.1 

.30 

T    7 

—     24.8 

lu.iO.vJ 

1  £*O  A    Cl 

1624.6 

40.33 

+3.38 

3.8 

3.8 

.29 

8 

+1599.4 

lOi 

>o  o 

4.2 

39.53 

3.0 

3.6 

.28 

9 

3222.6    , 

1  O. 

1  n> 

So.2 

)•)   W 

3.7 

38.74 

2.2 

3.4 

.27 

10    +4845.4 


+1623.4     54  37.96 


3281.4       893.3       954.26 


O  App.  Dec.  S'.  D  App.  Dec.  S.      (8+8') 

h          o     /    //  o       /     //  /     // 

3  +12  0  53.92—11  47  26.8+13  27.12— 

4  144.59  5529.9+  614.69 

5  235.24  12    329.6—  054.36 

6  325.86  1125.9      8   0.04 
TO  7  4 16.47  19  18.7     15    2.23 

8  5   7.05         27    8.0    22   0.95 

9  557.61          3453.7     2856.09 


Sin  28.  (a— a')2-  log*.  e. 

9.6019+5.2960— 1.3336— 21.56  ,  Q  97 

.6066  5.0475  1.0898  12.29+^—2.53, 

.6110  4.6977  0.7444  5.55  J'{|  2.62 

.6158  4.1024  0.1539  1.43  .  H,  2.69 

.6202  0.4557  6.5116  0.00 +}'  07  2.80 

.6243  4.0764  0.1364  1.37  i'«i  288 

.6286  4.6854  0.7497  5.62  ?„?— 2.96- 


10  +12  648.15— 12  4235.8—3547.65-4.6327+5.0400—1.1084—12.83 


208 


THEOKY   OF   ECLIPSES. 


190 


y- 

y'. 

logre'. 

For  TO. 

For  first  approx. 

3*  + 

828.7 

A  A]  7 

—433.0 

+3.21112 

XQ 

—1.39386 

a/ 

+3.21075 

4 

+ 

387.0 

'rtL.  1 

40K    Q 

429.7 

104 

2/o 

—2.95530 

y' 

—2.62377 

5 

48.8 

rrOO.o 
49Q  R 

426.7 

093 

tan  M      8.43856 

tan 

N      +0.58698 

6 

478.6 

'iZy.o 

49  Q  « 

423.6 

685 

M 

+181  34  21 

N 

+104  30  42 

7 

902.2 

^r^O.U 
,|-t  rr    y| 

420.5 

075 

cos 

9.99984 

sin 

9.98592 

8 

1319.6 

41/.4 

417.4 

064 

log 

m  +2.95546 

log 

1  :  n  +6.77517 

9 

1730.5 

Af\A  Q 

414.1 

051 

0  —2134.8 

—  4.U41.O 

—410.9 

+3.21043 

Example,  Preliminary  Work. — After  interpolating  a  and  a!  +  12A 
get  the  quantity  (« —  «')  in  time,  then  x  by  396.  By  retaining 
(a  —  a')  in  time,  it  is  reduced  to  arc  by  the  logarithm  of  15  when 
adding  the  other  logarithms.  The  hourly  variation  is  similar  to  that 
in  solar  eclipses  (Art.  30),  the  epoch  hour  T0  being  first  assumed  near  the 
middle  of  the  eclipse,  and  the  hourly  motions  of  x  then  computed. 
TT  may  be  reduced  to  the  latitude  of  45°  by  means  of  the  subjoined 
table.  The  correction  is  always  to  be  deducted 
numerically,  and  the  simplest  method  of  applying 
it  is  to  deduct  it  from  the  sun's  parallax  TT',  which 
is  constant,  and  apply  this  to  TT,  giving  at  once 


Reduction  of  IT 

to  45°. 

77. 

cor. 

52" 

—  5".l 

53 

5  .2 

54 

5  .3 

55 

5  .4 

56 

5  .5 

57 

5  .6 

58 

5  .7 

59 

5  .8 

60 

5  .9 

61 

6  .0 

62 

—6  .1 

The  two  semidiameters  are  to  have  the  constants 
of  irradiation  deducted  as  in  solar  eclipses,  Art. 
21. 

The  small  term  e  is  conveniently  computed  by 
formula  (395),  since  log  sin  2£  can  be  gotten  men- 
tally to  four-place  decimals,  (a  —  a')  is  given 
above,  and  simply  doubled  for  the  square,  and 
the  constant  given  by  its  logarithms  is  composed 
of  the  constants  in  CHAUVENET'S  form — viz., 

(i  sin  I"  x  15)2 

sin  V 

This  term  is  to  be  algebraically  subtracted  from  S  +  8f.  Proceed 
with  the  formula,  getting  y,  and  thus  the  hourly  motions  yf.  If  y 
changes  signs,  it  indicates  a  large  eclipse.  The  logarithm  of  x'  may 
be  conveniently  gotten,  but  y'  must  be  retained  in  numbers,  since  its 
logarithm  cannot  be  easily  interpolated. 

All  these  quantities  should  be  differenced,  but  these  are  omitted  in 
the  example  for  want  of  space. 

The  quantities  mM,  nN,  for  the  epoch  hour  are  next  required  ;  the 
two  former  are  computed  but  once  for  the  eclipse,  but  the  latter  are 
for  the  first  approximation  only  (formulae  (398),  (399)). 

Elements. — These  may  now  be  gotten  from  the  preceding  data,  and 
are  similar  in  all  respects  to  those  for  solar  eclipses  (Art.  21),  except 


190 


LUNAR   ECLIPSES. 


209 


that  we  have  here  opposition  instead  of  conjunction  ;  the  formulae  for 
these  are  the  same. 

191.  The  Times  y  Angles  of  Position,  Magnitude.  —  For  the  times 
two  approximations  are  required,  taking  for  the  first  the  several 
quantities  for  the  epoch  hour  or  an  hour  near  the  middle  of  the 
eclipse,  and  computing  in  three  columns  instead  of  six,  as  in  the 
second  approximation  ;  but  after  the  angle  <p  is  reached,  there  will 
then  be  six  columns  on  account  of  the  two  values  of  cos  (p.  This 
first  approximation  is  omitted  in  the  example,  but  the  resulting  times 
are  placed  at  the  head  of  the  columns,  for  which  time  the  quantities 
are  to  be  taken  from  the  eclipse  data  above  computed.  If  a  total 
eclipse  of  eighteen  years  ago  has  become  partial,  it  will  be  found  that 


COMPUTATION  FOR  THE  TIMES. 


Penumbra. 
3A48'n.95  9*56^.60 


Shadow. 
5*  0™.20  8h  45"*.35 


Totality. 
10W.09  7*  35m.47 


(393)  TT,  -f  TT' 


2 
1:50 

3V(7rl  +  *  ±  s/ 
S 

L 


3286.4 
+954.3 

4240.7 
84.8 

4325.5 
+894.6 

5220.1 


3281.4 
+954.3 

4235.7 
84.7 

4320.4 
+893.2 

5213.6 


3285.4 
—954.3 

2331.1 
46.6 

2377.7 
+894.3 

3272.0 


3282.4 
—954.3 

2328.1 
46.6 

2374.7 
+893.4 

3268.1 


3284.5 

—954.3 

2330.2 

46.6 

2376.8 
—894.1 

1482.7 


3283.3 
—954.3 

2329.0 
46.6 

2375.6 
—893.7 

1481.9 


(399)  V 

2/o' 

tanJV 
N 
sin 
log  1 :  n 

(400)3/—JV 
sin  (M — 
cos(M- 
m  sin  (M 
logL 


+3.21106  21043  21093  21054  21083  +3.21068 

—2.63377  61384  63012  61794  62644  —2.62180 

0.57729  59659  58081  59260  5S439  0.58888 

+104  49  29  104  12  25  104  42  36  104  19  58  104  35  40+104  27  3 

9.98530  98651  98552  98627  98575  9.98604 

+6.77424  77608  77459  77573  78492  +6.77536 


+76  44  52 
N)  +9.98828 
N)  +9.36029 
—  JV)+2.94374 
+3.71768 
+9.22606 


77  21  56 
98936 
33991 
94482 
71714 
22768 


$ 

cos  V 

(401)  log  (1) 

(2) 

Nos.  (1) 
-(2) 

T 

(402)  T 

(404)  Middle 
14 


76  51  45 
98848 
35657 
94394 
51481 
42913 
+164  25 


77  14  23 
98914 
34415 
94460 
51429 
43031 

+15  38 


76  58  41  +77  7  18 
98868  +9.98894 
35280  +9.34807 
94414  +9.94440 
17105  +9.17082 
77309  +9.77358 


—9.99377  +9.99372  —9.98373  +9.98364  —9.90589  +9.90562 

—0.48569  +0.48694  —0.27313  +0.27366  —9.85186  +9.85180 

+9.08999      9.07145      9.08662      9.07534      9.08318  +9.07889 

—3.0598     +3.0686    —1.8755     +1.8778    —0.7110  +0.7109 

—0.1230    —0.1179    —0.1221     —0.1190    —0.1211  —0.1199 

—3.1828    +2.9507    —1.9976     +1.7588    —0.8321  +0.5910 

3.8172        9.9507        5.0024        8.7588        6.1679  7.5910 

3  49.03      9  57.04       5  0.14         8  45.53       6  10.07  7  3.546 


=  - 0.120 


T=  6.880  =  6^52^.80 


210  THEORY  OF   ECLIPSES.  191 

Angles  of  Position.  (406,  407)  Magnitude. 

(405)  Constant    +  180°    0'  +  180°    0'  A  Mean  value,  log  2.94427     Nos.  879.6 

N  +104    43  +104    20  M_L  —  A  =  3270.0—879.6  =  «  OQQ 

$  +164    25  +    15    38  2s  1787.8 

+  89      8       299    58 

Q  First  89  to  E.        Second  60  to  W. 

The  Moon  in  the  Zenith. 

First  Contact.  Last  Contact. 

(408)  T                             5*  0»14  8*  45m.53 

E                              +  1  .41  +1  .44 

Constant                  12    0  12      0 

Sum                         17    1  .55  20    46  .97 

—  (a  —  a7)                      +3  .72  —3  .40 

Sum                         17    5  .27  20    43  .57 

256°  18'  W.  310°  W  W. 


(  r         , 
.(marc) 

(409)  0  =  d  12°    4'.  S.  12°  33'.  S. 

in  the  two  columns  headed  totality,  sin  <p  will  result  greater  than 
unity,  and  the  work  can  therefore  be  carried  no  further.  And  in 
general  throughout  this  whole  theory  of  solar  and  lunar  eclipses, 
when  this  results  there  is  no  eclipse. 

There  seems  to  be  nothing  difficult  about  this  computation,  espe- 
cially to  one  familiar  with  solar  eclipses.  One-fiftieth  of  the  (^  -f- 
TT'  =h  s')  is  added  on  account  of  the  earth's  atmosphere,  as  above  re- 
marked. The  angle  <p  is  required  only  for  the  shadow  for  use  with  the 
angles  of  position.  The  cosine  is  negative  for  beginning  and  positive 
for  ending,  which  gives  the  quadrant  for  tp  and  its  sign  results  from 
the  computation.  The  final  times  should  not  differ  more  than  a  few 
tenths  of  a  minute  from  those  of  the  first  approximation,  and  usually 
the  greatest  difference  is  in  the  columns  for  penumbra,  and  the  least 
differences  for  totality. 

In  the  first  approximation  the  times  are  distributed  symmetrically 
about  the  middle  time,  as  shown  by  differencing  them  ;  but  in  the 
second  they  are  not,  on  account  of  the  changes  in  all  the  semi- 
diameters,  parallaxes,  etc. 

Professor  CHATJVENET  has  not  computed  his  example  as  accu- 
rately as  it  should  have  been,  making  but  one  approximation, 
and  computing  x  and  y  at  intervals  of  three  hours.  It  seems  also 
that  the  English,  French,  and  German  Nautical  Almanacs  are  not  so 
accurate  as  we  are  in  this  respect,  since  their  times  are  symmetrical 
about  the  middle  times,  indicating  but  one  approximation  for  the  times. 

The  middle  of  the  eclipse  is  already  given  in  the  computation  for 


191  LUNAR   ECLIPSES.  211 

totality,  or  in  a  partial  eclipse  for  shadow.  The  mean  of  the  two 
values  for  beginning  and  ending  should  be  taken  to  give  the  time  of 
the  middle. 

For  the  angles  of  position  and  magnitude  the  signs  must  be 
regarded  as  also  in  the  foregoing  work.  For  a  partial  eclipse,  as 
above  noted,  the  magnitude  is  less  than  unity.  In  the  present 
example  the  moon  is  wholly  obscured  (unity),  and  its  nearest  limb 
0.338ths  of  its  diameter  within  the  cone  of  shadow. 

For  the  moon  in  the  zenith,  it  is  more  convenient  to  reduce  both 
(a  —  a')  and  the  equation  of  time  to  minutes  and  hundredths  before 
interpolating.  The  former  is  a  fraction  of  1  hour  and  the  latter  a 
fraction  of  24A  to  the  time  T  for  beginning  and  ending  of  shadow. 
(a  —  a'}  will  always  be  -f  for  beginning  and  —  for  ending,  E  is 
to  be  taken  from  page  II.  for  the  month  in  the  Nautical  Almanac, 
which  reduces  mean  to  apparent  time.  Finally,  reduce  the  longitudes 
from  time  to  arc. 

192.  Lunar  Appulse. — This  is  the  name  given  to  a  very  close 
approach  of  the  moon's  limb  to  the  earth's  shadow  without  enter- 
ing it.     An   occurrence   of  this  kind  took  place  in  1890,  June  2. 
The  quantity  (^  -f  nf  —  sf)  was  about  2642".    (I  have  not,  however, 
the  original  figures  to  refer  to.)     One-fiftieth  of  this,  53",  and  the 
nearest  approach  of  the  limb  of  the  moon  to  the  shadow,  was  19".3 ; 
but  this  latter  quantity  is  not  given  in  the  Nautical  Almanac.     The 
value  53"  is  empirical  and  doubtful,  so  that  I  inserted  the  following 
clause  :  "  The  nearness  of  the  approach  and  the  uncertainty  of  the 
effect  of  the  earth's  atmosphere,  render  it  doubtful  whether  the  moon 
will  enter  the  shadow  of  the  earth  or  not." 

A  still  more  doubtful  case  occurred  on  December  llth  of  the  same 
year,  in  which  the  computation  resulted  in  an  eclipse ;  but  the  dura- 
tion of  the  partial  eclipse  was  but  3m  5*.  5,  and  the  magnitude  0.005, 
which  is  equivalent  to  but  9 ".3.  Also  in  1900,  June  12,  the  mag- 
nitude was  0.001,  equivalent  to  but  1".9. 

The  analytical  condition  for  an  appulse  is  that  the  magnitude, 
equation  (407),  is  negative  or 

A  >  L  (410) 

that  is  the  distance  of  the  moon's  limb  is  greater  than  the  radius  of 
the  shadow. 

193.  Graphic  Method. — In  the  following  manner  a  lunar  eclipse 
may  be  projected,  giving  the  times  to  one  or  two  minutes.     It  is  re- 


212  THEORY   OF   ECLIPSES.  193 

marked  first,  that  as  we  see  the  moon  in  the  heavens,  its  motion  is 
from  the  right  hand  toward  the  left,  which  is  the  positive  direction 
in  this  method  ;  and  the  positive  direction  for  angles  is  from  the 
north  toward  the  left  hand.  We  will  take  the  eclipse  previously 
computed,  1902,  April  22. 

A  convenient  scale  for  this  projection  is  1000"  to  two-thirds  of  an 
inch,  employed  by  the  author  for  all  the  lunar  eclipses  he  has  com- 
puted. Fig.  27  is  drawn  to  a  scale  of  1000"  to  one  inch,  but  re- 
duced in  the  figure  one  half,  or  2000"  to  one  inch. 

First  draw  the  axes  of  X  and  Y  at  right  angles  to  one  another, 
Fig.  27,  Plate  X.,  and  lay  off  the  negative  values  of  x  on  the  right, 
and  the  positive  values  on  the  left  of  the  origin;  on  these  points 
erect  the  ordinates  :  the  positive  above  and  the  negative  below  (Art. 
190).  The  shadow  is  a  circle  drawn  from  the  origin  of  coordinates  0 
with  the  radius  (Art.  191). 

+  *'  —  a'   =  2376" 


and  the  penumbra  a  circle  with  the  radius 

*'  +  a'    =  4323" 


which  are  mean  values  of  those  previously  computed. 

A  line  drawn  through  the  ends  of  the  ordinates  is  the  path  of  the 
centre  of  the  moon  through  the  shadow,  the  ordinates  marking  the 
integral  hours.  With  the  dividers  opened  to  the  moon's  semidiam- 
eter,  s  =  894"  move  one  leg  along  the  path  until  the  other  leg  is 
just  tangent  to  the  penumbral  circle  on  the  outside  ;  these  two  points 
will  be  the  first  and  last  contacts  of  the  moon  with  the  penumbra. 
Likewise  with  the  same  radius,  two  circles  drawn  tangent  to  the 
shadow  on  the  outside,  will  show  the  beginning  and  ending  of  the 
partial  eclipse  ;  and  two  other  circles  tangent  to  the  shadow  on  the 
inside  will  show  the  beginning  and  ending  of  totality.  If  a  line  be 
drawn  from  the  origin  of  coordinate  perpendicular  to  the  path,  it  will 
cross  the  path  at  a  point  J,  which  marks  the  middle  of  the  eclipse. 
Describe  a  circle  from  this  point  with  the  same  radius  as  the  others. 

The  moon's  path  may  now  be  divided  up  into  10-minute  spaces, 
or  even  closer  if  the  scale  permits,  and  the  centres  of  the  small 
circles  will  mark  the  times  of  the  phenomena.  The  point  K,  where 
the  path  crosses  the  axis  of  Y,  is  the  time  of  opposition  in  right 
ascension. 

The  radii  used  above  are  the  true  radii  of  the  shadow  and  penum- 
bra, these  quantities  ±  the  moon's  semidiameter  s,  used  in  the  com- 


193  LUNAR   ECLIPSES.  213 

putation,  give  the  centres  of  the  moon,  which  are  necessary  to  find 
the  times  there,  as  well  as  in  this  graphic  method.  In  case  x  and  y 
have  not  already  been  computed,  this  projection  may  be  made  by  the 
method  for  solar  eclipses,  Art.  161. 

194.  The  various  quantities  employed  in  the  general  formulae  may 
also  be  shown  in  this  projection.  We  will  briefly  recapitulate  some 
of  them  ;  x  and  y  have  already  been  plotted  as  above.  The  hourly 
motions  are  the  distance  between  abscissas  and  the  differences  of  the 
ordinates,  or  very  nearly  so  ;  and  being  mean  hourly  changes  cannot 
be  easily  shown.  But  neglecting  small  terms,  we  have  at  the  9-hour 
ordinate  erected  from  the  point  A,  drawing  9  B  to  the  10-hour  ordi- 
nate,  and  parallel  to  the  axis  of  Xy  SB  =  x',  and  B  10  —  y'. 

Also  the  7-hour  point  being  Tw  we  have  — 


OTC=(M-N) 
OI=msm(M—N) 

Connecting  the  centres  of  the  moon  C  C',  with  the  centre  of  the 
shadow  0  in  the  formulae. 


OC=L 

C4  =  t  =  164°  25'         0  O  K=  $  =  15°  38' 


mcos  (M—N) 

and  dividing  these  latter  by  n  (the  motion  of  the  shadow  in  one 
hour),  we  have  the  times  of  describing  these  distances,  and  hence  r 
and  T  follow. 

I  C  and  I  Cf  are  here  considered  as  equal,  but  the  changing  of  all 
the  quantities  during  the  eclipse  causes  them  to  differ  slightly,  by  an 
amount  too  small  to  be  seen  on  a  drawing.  Also  x'  yr  n  are  here 
shown  as  the  motions  between  two  hours  ;  in  strictness,  they  are  the 
motions  at  the  given  hour. 

From  the  centres  of  the  moon  at  first  and  last  contacts  erect  the 
lines  CP  and  O'P'  parallel  to  the  axis  of  F,  then 

Angles  of  Position  j  PCO    =  89°    }  Measured  toward  the 
1  P'  Cf  0  =  300°  I      East  or  left  hand. 


214  THEORY   OF   ECLIPSES.  194 

For  the  magnitude 

OR  =  L,  for  totality. 

01=  m  sin  (M  —  N)  =  J,  for  middle  of  the  eclipse. 
IE  =  L  -  A 

This  latter,  divided  by  the  moon's  diameter,  gives  the  magnitude.  It 
is  seen  that  if  L  —  J,  the  path  must  be  so  far  moved  out  that  I  falls 
upon  R,  the  moon  will  be  just  tangent  to  the  shadow  on  the  outside, 
and  M  =  0.  And  if  the  moon  is  tangent  to  the  shadow  on  the  inside, 
the  path  at  I  is  then  distant  from  R  by  just  2s,  so  that  M=  unity, 
the  limit  between  a  partial  and  total  eclipse. 


SECTION    XXV. 

TRANSITS  OF  MERCURY  AND  VENUS  ACROSS  THE  SUN'S  DISK. 

195.  Data,  Elements. — The  formulae  for  this  phenomenon  are 
derived  from  the  fundamental  formulse  for  solar  eclipses,  with  such 
modifications  as  the  circumstances  require,  only  the  results  being 
given  in  the  present  work.  The  data  required  from  the  Nautical 
Almanac  are  the  right  ascensions,  declinations,  parallaxes,  and  semi- 
diameters  of  both  the  sun  and  planet  to  be  interpolated  for  three  or 
four  hours  before  and  after  the  time  of  inferior  conjunction.  Also 
the  sidereal  time  and  equation  of  time  from  the  Almanac,  page  II., 
for  the  mouth,  and  the  distance  of  the  planet  from  the  earth  at  the 
time  of  inferior  conjunction  to  compute  or  to  verify  the  semidiam- 
eters  and  parallax.  These  latter,  since  the  year  1900,  are  now  given 
to  two  decimals  for  all  planets,  but  formerly  were  given  to  only  one 
decimal.  The  distance  of  the  earth  from  the  sun,  log  rf,  is  required 
only  to  check  the  semidiameter,  from  which  the  constant  of  irradia- 
tion must  be  deducted. 

The  elements  are  given  in  the  Nautical  Almanac,  similarly  to  those 
for  eclipses  (Art.  21). 

The  constant  k  is,  as  usual,  the  ratio  of  planet's  semidiameter 
divided  by  that  of  the  earth.  The  adopted  semidiameters  given  in 
the  Nautical  Almanac  for  a  number  of  years  past  are 

Mercury,  3".34  Venus,  8.546  (411) 

at  their  mean  distance  from  the  sun,  which  is  unity  for  each.     The 


195  TRANSITS   OF   MERCURY   AND   VENUS.  215 

earth's  semidiameter  at  the  mean  distance,  which  is  unity,  is  the 
sun's  parallax  TZ'  . 

The  values  of  Jc  given  by  Professor  CHAUVENET  ;  viz.  : 

For  Mercury,  0.3897  For  Venus,  0.9975          (412) 

depend  upon  ENCKE'S  parallax,  which  he  uses  throughout  his  chapter 
on  Eclipses. 

For  the  transit  of  Mercury,  1894,  Nov.  10,  the  author  of  these 
pages  used  NEWCOMB'S  parallax,  8.848,  which  gave 

For  Mercury,  0.37748  (413) 

For  the  approaching  transit  of  Mercury,  1907,  Nov.  14,  will 
undoubtedly  be  used  the  constant  8.800  of  the  Paris  Conference, 

which  gives 

For  Mercury,  Jc  =  0.37954,  log  9.57926  (414) 

This  constant  k  is  not  used  in  the  computations  of  transits  as  it  is 
in  eclipses,  on  account  of  the  modifications  made  in  the  formulae  ;  it 
is,  however,  useful  for  verifying  the  semidiameter  and  parallax. 

In  equations  of  condition  the  constant  k  occurs,  and  various 
changes  in  the  formulae  are  necessary,  but  for  this  the  reader  is 
referred  to  CHAUVENET'S  own  words,  since  this  branch  of  the  sub- 
ject, both  in  eclipses  and  in  transits,  has  not  been  considered  in  the 
present  work. 

196.  General  Formulce.  —  The  notation  is  the  same  as  in  eclipses, 
except  where  otherwise  stated  ;  a,  S,  s,  and  other  quantities,  which  in 
solar  eclipses  represent  the  moon,  are  here  used  to  denote  the  planet, 
which,  like  the  moon,  comes  between  the  earth  and  sun. 

I.  For  the  centre  of  the  earth.     (CHATJVENET,  Arts.  356-58.) 

m  sin  M  =  (a  —  a')  cos  }  (d  +  <?')  j 
<5  —  <?'  j 


Ja',  Ja,  Ad',  and  Jd,  being  the  hourly  motions, 

n  sin  N=  (Aa  —  Aa'}  cos  I  (3  +  #0 


.  msin(M-N) 

=  - 


.                             -  //117\ 

sin  ^  =  -  (417) 

s'is 

In  parts  of  an  hour,       r  =  -  cos  0  --  cos  (M—  N)  (418) 

n  n 

T=T0  +  r  (419) 

(420) 


216  THEORY  OF   ECLIPSES.  196 

In  the  above  formulae  s'  ±  s  is  the  distance  of  the  centres  of  the  sun 
and  planet,  the  upper  sign  for  external  contacts  and  the  lower  for 
internal.  $  is  obtuse  for  ingress  and  acute  for  egress,  as  in  eclipses. 
T0  is  the  epoch  hour  and  r  the  correction  for  it.  Q  is  the  angle  of 
position  of  the  point  of  contact,  the  two  values  of  (p  giving  two  values 
for§. 

If  h  represents  3600,  the  number  of  seconds  in  one  hour,  the 
above  becomes 

In  seconds,    T  =  TQ  -f  h  (^^]  cos  <1>  -  —  cos  (H  -  JV)  *  (421) 

\     n    /  n 

We  also  have,  as  in  solar  eclipses,  for  the  distance  and  time  of 
nearest  approach  of  the  centres, 

T,  =  To  -  —  cos  (M  -  N)  (422) 

n 

II.  For  any  given  place.  By  a  method  devised  by  LAGRANGE, 
which  CHAUVENET  gives  as  new,  the  times  for  any  place  are  given 
by  the  following  formula  : 

T'  —  T  +  *"**  IP  cos  <pr  sin  D  +  p  cos  ?'  cos  D  cos  (0  —  w)]  (424) 
n  cos  ^ 

in  which  Tj  n,  (p,  D,  #,  TT  —  nf  are  all  constant,  and  given  in  the 
computation  for  the  centre  of  the  earth.  T  is  the  time  computed 
for  the  centre  of  the  earth,  and  p  sin  ^',  p  cos  (pf  the  geocentric 
positions  of  the  given  place  as  used  in  eclipses  (Art.  149). 

For  these  constants  compute  as  follows,  taking  the  quantities  from 
the  computation  for  the  centre  of  the  earth  for  the  proper  times. 
For  the  Nautical  Almanac  these  are  computed  but  twice  for  ingress 
and  egress  at  exterior  contacts. 

Q  =  N+</>  (425) 

I      f 
tan  r  =  -  ^—f  tan  J  (s'  ±  s)  (426) 

/sin  F  =  sin  f  \ 

f  cos  F  =  cos  Y  cos  Q  ) 
cos  D  sin  [J.  —  £  (a  -}-  a')]  =  cos  Y  sin  Q  "| 

cos  D  cos  \A  -  J  (a  +  a')]  =  -  /  sin  [  J  (<?  +  *')  +  JF]  >    (428) 
sin  D  =  f  cos  p  (5 "+  5')  4-  F~\       ) 

*  CHAUVENET  has  a  misprint  here  on  page  598 — the  sign  following  T0  should 
be  +,  not  —  line  24. 


196  TRANSITS   OF   MERCURY   AND   VENUS.  217 

Then  the  constants  are 

9  =  P.— A  (429) 

B  =  *""*'  h  sin  D  (430) 

n  cos  ^ 

0  =  ^-— -  A  cos  D  (431) 

?&  COS  0 

And  equation  (424)  becomes 

T'  =  T  -f-  Bp  sin  ?'  -f  Cp  cos  ?>'  cos  (0  —  o>)  (432) 

For  the  sun  in  the  zenith  (Art.  189), 

^lJ,'+-E~(a~a')}  (433) 

197.  Example. — The  example  here  given  is  from  the  author's  com- 
putation sheets  of  the  Transit  of  Mercury,  1894,  November  10.  The 
preliminary  work,  data,  from  the  Almanac  elements,  etc.,  are  omitted. 
This  omission  will  not  interfere  with  the  understanding  of  the  exam- 
ple or  formulae,  since  the  data  are  so  designated  in  the  margin  of 
the  example. 

As  in  eclipses,  the  work  is  kept  compact,  and  quantities  seldom 
rewritten  which  are  previously  given.  In  this  respect  he  has  acted 
upon  a  casual  remark  of  one  of  his  predecessors  in  several  of  the 
different  portions  of  the  Nautical  Almanac — a  gentleman  known 
throughout  the  United  States  and  Europe — "  I  never  repeat  a  figure 
when  it  can  be  avoided." 

In  the  first  approximation,  which  is  omitted  from  the  example,  the 
quantities  are  taken  out  of  the  tables  of  data  for  an  hour  near  the 
middle  of  the  transit.  For  this  time,  which  is  arbitrarily  selected, 
it  is  convenient  to  select  the  exact  instant  of  conjunction,  T0,  for 
which  time  M  =  0,  m  =  d  —  df,  a  =  af,  and  most  of  the  other  quan- 
tities are  already  gotten  for  the  elements.  The  computation  thus  con- 
sists of  but  one  column  until  (p  is  reached,  when  the  two  values, 
s1  -j-  8  for  external  contacts,  and  s'  —  s  for  internal,  give  two  quite 
different  values  for  sin  ^,  and  each  of  these  will  give  two  values  for  </>, 
the  obtuse  for  ingress,  and  the  acute  for  egress,  so  that  the  first 
approximation  now  consists  of  four  columns,  as  in  the  remainder 
of  the  work.  The  resulting  times  are  given  at  the  head  of  the  four 
columns  of  the  example  ;  and  for  these  times  the  quantities  are  taken 
from  the  tables  of  data  for  the  final  results.  These  times  now  be- 
come TQ  in  each  of  the  four  columns,  and  are  so  noted  in  the  margin. 
I  have  discarded  CHATJVENET'S  contractions,  writing  cr0  for  J  (a  -f  a'), 
and  restored  the  original  quantities,  making  the  formulae  clearer, 
though  a  little  longer. 


218 


THEORY   OF   ECLIPSES. 


197 


TRANSIT  OF  MERCURY,  1894,  Nov.  10. 

I.  For  the  Centre  of  the  Earth  G.  M.  T. 


From  1st  Approx. 

TO 

3*  55m  298.3  9A  12m  78.7 

3ft  57m  13S.3  9A  10W  23S.7 

Data 

s' 

969.80         969.85 

969.80         969.85 

Data 

s 

+4.94 

—4.94 

s'±s 

974.74         974.79 

964.86         964.91 

Data 

Aa 

—187.20     —186.62 

Ad         +105.12     +105.17 

Data 

Aa' 

+151.80      +151.91 

Ad'         —41.85       —41.68 

( 

(AJ—  } 

(Aa  —  Aa') 

—339.00     —  338.53J 

Data  (415) 

a  —  a' 

+1009.81     —778.09 

+1000.01    .—768.31 

Data 

6  —  d' 

—144.81      +630.54 

—140.57      +626.30 

Data 

»(*+*0 

—17°  18'  6"  17°  15'  19" 

17°  18'  5"—  17°  15'  20" 

log  (a  -a') 

+3.00424    —2.89103 

+3.00000   —2.88554 

cos  £  (8  +  6r 

)    +9.97989    +9.98000 

+9.97989    +9.98000 

(415) 

m  sin  M 

+2.98413   —2.87103 

+2.97989    —2.86554 

m  cosM 

—2.16080    +2.79971 

—  2.14790    +2.79678 

tan  If 

0.82333       0.07132 

0.83199       0.06876 

M 

+98  32  31    —49  41  0 

+98  22  33    —49  31  0 

sin  M 

9.99516       9.88223 

999534       9.88115 

logm 

+2.98897       2.98880 

2.98455   +2.98439 

(416)  log(Aa-Aa') —2.53020  —2.52960 
rasing  +2.51009  +2.50960 
nvosN  2.16723  2.16688 
tan  N  0.34286  0.34272 
JV  _65  34  40  —65  34  14 
sin  N  9.95929  9.95926 
log(l:n)  +7.44920  +7.44966 

(417)  M—  N  +164711+155314 
dn(M-N)    +9.43716    +9.43734 
cos(^f—  N)    —9.98310    +9.98308 

Least  distance       msin(M-N)  +2.42613  +2.42614 

s/  +  s               +2.98889  2.98891 

sinV                  +9.43724  9.43723 

V                       +164  7  0  +15  52  59 

(418)  cosV                —9.98309  +9.98309 
log  (1)  X  3600  —3.97748  +3.97796 
log  (2)  X  3600  —3.97757  +3.97784 

— Nos.  (2)          +2  38  16.6  —2  38  22.5 
6  33  45.9     6  33  45.2 
Nos.  (1)          —2  38  14.7  +2  38  25.2 
T  3  55  31.2     9  12  10.4 


—65  34  40  —65  34  14 

+7.44920  +7.44967 


(422)  Middle    = 

Final  times. 
Contacts. 


+163  57  13 

+9.44156 

—9.98274 

+2.42611 

-s     2.98447 

9.44164 

+163  57  3 

—9.98273 

—3.97270 

—3.97279 

+2  36  32.8 

6  33  46.1 

—2  36  30.7 

3  57  15.4 


+16  3  14 

+9.44176 
+9.98272 
+2.42615 
+2.98449 
+9.44166 
+16  3  0 
+9.98273 
+3.97319 
+3.97307 

—2  36  38.7 
6  33  45.0 

+23641.4 
9  10  26.4 


I.  Ingress.     IV.  Egress.     II.  Ingress.     III.  Egress. 


(422)  Middle  for  Nautical  Almanac  from  1st  Approx.  6A  33™  488.5 

(423)  Least  distance  of  centres,  m  sin  (M—  N)  4    26  .76 


197 


TRANSITS   OF   MERCURY   AND   VENUS. 


219 


TRANSIT  OF  MERCURY,  1894,  Nov.  10. 

Constants. 

T  3A  55m  3P.2  9A  12™  10«.4 

+  TT'     =  13".08  +    8'.94  22".0 

—  TT'         =  4    .14 

Data  at  tf    {    g,  _^  g      =   4  .94  +  15    9//tg3  =  16  14  .77 
«'      «=  8    7  .38 


log  TT  +  TT'          1.34282 
tanj(s'+s)      7.37343 
8.71625 

rr  —  TT'  0.61700 

tan  7  8.09925 

sin  7  +8.09922 

cos  7  +9.99997 

(425)^  =  ^+^+98  32  20    —49  41  15 
sin  Q  +9.99516      —9.88226 

cos  Q  —9.17167       +9.81087 


(427)  /  sin  F  +8.09922 

/  cos  F  —9.17164 

tan  F  8.92758 

F  +175  9  43 

cos  9.99845 

log/  +9.17319 

(42S)l(6+6')+F  +157  5137 

sin  +9.57619 
cos 

cos  D  sin  ( 
cos  D  cos  ( 
tan(  ) 


sn 

cosD 

sinD 


—9.96674 
+9.99513 
—8.74938 

1.24575 
+93  15  1 

9.99930 
+9.99583 
—9.13993 


+8.09922 

9.81084 

8.28838 

+16  46 

9.99992 

9.81092 

—16833 

—9.44409 

+9.98253 

—9.88223 

9.25501 

0.62722 

—76  43  30 

9.98824 

+9.89399 

49.79345 


T 

Contacts. 

(430)  (TT  —  TT')  X  3600 
w  cos  V7 


(431)  log  C 

f  ft  (noon) 

SecTesM  Jd.  interval 

(^  ft  (time) 

(429)  11  (arc) 

}  (a  +  a') 


(433)  3" 
-& 

-(a -a') 


3*  55TO    31S.2 
I.  Ingress. 

+4.17330 
—2.53389 
1.63941 
+0.77934 

—1.63524 
15*  18OT  3P.4 

3  55    31 .2 

38.7 
19    14    41.3 

288°  40'  19".5 
225     57      3 
+93     15      1 
319     12      4 
329    28    15 
Sun  in  the  Zenith. 
3*  55". 52 
+  15  .92 

4  11  .44 
—1  .13 

4    10  .31 

35'  W. 


(433) 0 

Q 

(432) 


62C 
— 17C 


17' S. 


9A   12W  10'.4 
IV.  Egress. 

+4.17330 

+2.53343 

1.63987 

+1.43332 

+1.53386 

15*  18™  3P.4 

9    12    10 .4 

1     30.7 

0    32    12.5 

go    3/    7// 

225  55  30 

—76  43  30 

149  12  0 

218  51  7 

9*  12"M7 

+15  .90 

9    28  .07 

+0  .86 

9     28  .93 

+142°  14' W. 
—17°  21'  S. 


Angles  of  Position. 
98°  32'  to  E. 


—49°  41'  to  W. 
Final  Equations  for  the  Times  for  any  Place. 
=  3*  55™  31'.2  +  [0.77934]  p  sin  $'  —  [1.63524]  p  cos  0' 

cos  (329°  28'  15"  —  w) 

=9   12    10  .4  +  [1.43332]  p  sin  0' +  [1.53386]  p  cos  tf 
cos  (218°  51 '7"  —  u) 


220  THEOKY   OF   ECLIPSES.  197 

Owing  to  the  retrograde  motion  of  Mercury  at  conjunction  infe- 
rior, a  —  a'  has  a  contrary  sign  to  that  in  eclipses,  being  +  for 
beginning  and  —  for  ending ;  and  the  planet  is  seen  passing  over  the 
sun's  disk  from  left  to  right.  The  changes  in  the  hourly  motions 
are  so  small  that  the  quantities  for  the  I.  and  II.  contacts  are  the 
same  ;  likewise  those  for  the  III.  and  IV.  In  the  computations  the 
group  beginning  with  J  a  belongs  as  well  to  the  two  last  columns  as 
to  the  first.  A  d  also  belongs  to  the  two  first  columns,  they  are  so 
placed  merely  to  economize  space.  N,  Avhich  follows,  is  computed 
once  for  beginning  and  once  for  ending. 

The  terms  (1)  and  (2)  are  multiplied  by  3600,  by  its  logarithms, 
while  adding  the  other  logarithms,  giving  the  quantities  in  seconds, 
which  are  then  reduced  to  hours  and  minutes.  Each  column  gives 
a  value  for  the  middle — the  second  term ;  but  the  more  accurate 
value  is  that  derived  from  the  first  approximation,  which  is  made  for 
a  time  not  far  from  the  true  middle. 

A  partial  check  upon  the  computation  may  be  had  from  the  dis- 
tances between  the  centres  of  the  sun  and  planet,  so  that  we  have 
at  the  instant  of  contact — 

m  —  sf  +  s     or     m  =  sf  —  s  (434) 

Thus  in  the  present  example  for  the  four  contacts : 

I.  IV.  II. 

m  974.92  974.54  965.05 

s'±s  974.74  974.79  964.86 

Discrepancy  +18  —0.25  +0.19 

The  discrepancies  here  noticed  may  arise  partly  from  the  formulae 
for  m  being  only  approximate  (Arts.  127  and  173),  but  more  likely 
from  the  error  of  the  two  right  ascensions  when  reduced  from  time  to 
arc,  which  may  amount  to  0"15  in  each  (Arts.  19  and  29).  But  we 
may  notice  that  the  contacts  I.  and  II.  are  so  near  together  in  time 
that  they  would  usually  be  affected  by  the  same  error  and  with  the 
same  sign,  and  similarly  for  III.  and  IV. 

The  computation  for  the  constants  presents  nothing  especially 
difficult;  TTTT'  ss'  are  taken  from  the  elements  for  the  time  of  con- 
junction. They  are  almost  constant  during  the  continuance  of  the 
transit.  Quantities  required  here,  which  are  given  in  the  former 
part  of  the  work,  are  generally  not  repeated,  for  example,  N  and  </> 
are  given  above ;  but  Q  only  is  set  down  here.  The  work  thus 
stands  exactly  as  on  my  computing  sheets,  except  the  computation 


197  TRANSITS   OF   MERCURY   AND   VENUS.  221 

for  the  sidereal  time,  which  I  have  enlarged  a  little,  by  inserting  the 
mean  time  T,  the  quantity  J-  (a  -f  a')  from  the  data  and  that  which 
follows  it  from  the  previous  work.  The  sidereal  time  ft  is  the  sid- 
ereal time  at  noon  plus  the  mean  time  of  the  phenomenon  reduced 
to  sidereal  time,  a  and  a!  must  be  interpolated  separately  from  the 
data,  and  the  mean  \  (a  -+-  a')  gotten  ;  then  with  the  time  \_A  - 
J  (a  -f-  a')~\  from  the  first  computation,  A  is  found. 

The  sun  in  the  zenith  is  found  as  in  Lunar  Eclipses,  Art.  189. 
The  points  of  the'  earth's  surface  thus  found  give  the  hemisphere 
approximately  from  which  the  contact  may  be  seen,  and  in  a  stereo- 
graphic  projection  a  circle  described  from  this  point  with  the  radius 
of  the  sphere  will  show  the  hemisphere.  A  little  more  than  a  hem- 
isphere may  then  be  projected  stereographically. 

198.  Graphic  Representation. — This  is  not  of  as  much  value  as  in 
eclipses,  except  as  a  matter  of  instruction.     The  abscissas  and  ordi- 
nates  may  be  laid  off  from  formula  (415),  thus  : 

x  =  ?n  sin  M  =  (a  —  a')  cos  I  (d  -f-  <5r)  )  (435") 

y  =  wi  cos  Jf  =(*  —  *')  ) 

by  which  the  path  of  the  planet  may  be  drawn,  remembering  to  lay 
off  the  positive  values  of  x  on  the  left,  and  the  negative  values  on 
the  right,  of  the  origin  for  direct  vision,  since  the  motion  of  the  planet 
is  retrograde  at  inferior  conjunction.  The  positive  direction  for 
angles  is  from  the  north  point  toward  the  left.  The  radii  of  the  sun 
and  planet  are  their  semidiameters.  Mercury  will  be  a  mere  dot  on 
the  large  surface  of  the  sun. 

From  this  graphic  projection  the  various  quantities  used  in  the 
formulae  may  be  shown,  as  explained  in  Section  V.,  the  Extreme 
Times  Generally  in  Solar  Eclipses. 

The  only  charts  of  a  transit  that  have  been  given  in  the  Almanac 
are  those  for  Venus  in  1882.  They  differ  considerably  from  charts 
of  an  eclipse,  the  shadow  moving  from  east  to  west.  Limiting  curves 
would  rarely  occur,  and  the  rising  and  setting  curves  are  nearly 
great  circles  of  the  earth  intersecting  more  or  less  at  right  angles ; 
so  that  the  whole  transit  is  visible  to  about  one-fourth  of  the  earth, 
and  invisible  to  the  opposite  fourth. 

199.  Transits  of  Mercury  and  Venus  are  especially  employed  in 
determining  the  solar  parallax,  for  which  Venus  gives  much  more 
accurate  results  than  Mercury,  on  account  of  its  greater  proximity 


222  THEORY  OF  ECLIPSES.  199 

to  the  earth.  Professor  CHAUVENET  has  developed  this  method  in 
his  chapter,  already  quoted  from,  to  which  the  reader  is  referred. 

In  the  "Papers  relating  to  the  Transit  of  Venus  in  1874,  prepared 
under  the  direction  of  the  Commission  authorized  by  Congress/7  Part 
II.,  are  given  the  formulae  devised  by  Dr.  GEORGE  W.  HILL,  pre- 
viously referred  to  with  charts  of  the  transit. 

And  in  the  "  Instructions  for  Observing  the  Transit  of  Venus, 
December  6,  1882,"  are  similar  charts  prepared  by  Dr.  HILL.  The 
curves  are  similar  to  the  outline  curves  in  eclipses,  but  drawn  at  in- 
tervals of  one  minute.  There  is  also  given  an  orthographic  projec- 
tion of  the  earth  by  the  author  of  these  pages,  showing  the  path  of 
the  several  places  of  observation  over  the  sphere  during  the  continu- 
ance of  the  transit,  the  paths  and  circles  of  the  earth  projecting  into 
true  ellipses. 

In  Part  I.  of  the  "  Observations  on  the  Transit  of  Venus,  Decem- 
ber 8-9,  1874,"  by  Professor  NEWCOMB,  1880,  the  observations  of 
the  various  parties  of  observers  are  discussed.  There  is  also  given 
by  the  author  of  these  pages  a  description  and  detailed  drawings  of 
the  photoheliograph  used  for  reflecting  the  sun's  image  upon  the 
photographic  plate.  This  instrument  is  permanently  mounted  at 
the  Naval  Observatory. 

In  the  Astronomical  Papers  of  the  American  Ephemeris  and  Nauti- 
cal Almanac,  i.,  Part  VI.,  "  Discussion  and  Results  of  Observations  on 
the  Transits  of  Mercury,  1677  to  1881,"  Professor  NEWCOMB  has, 
among  other  things,  devoted  one  section  on  the  law  of  the  recurrence 
of  these  transits,  giving  diagrams  and  tables  of  the  transits  covering 
several  centuries. 


SECTION    XXVI. 

OCCULTATIONS  OF  FIXED  STABS  BY  THE  MOON. 

200.  General  Formuke  (Longitude). — The  occupation  of  a  fixed 
star  by  the  moon  is  but  a  special  case  of  a  solar  eclipse,  in  which  the 
sun  is  supposed  to  be  at  an  infinite  distance,  and  represented  by  the 
star,  its  parallax  and  semidiameter  are  zero  and  its  distance  r'  infin- 
ity. The  cone  of  shadow  then  becomes  a  cylinder,  whose  radius  is 
L  This  is  shown  by  equations  (36)  and  (38),  but  equation  (35)  takes 
the  indeterminate  form.  It  is  better  seen  in  the  equations  as  given 
in  Art.  182:  sin  /  is  zero,  its  cosine  unity,  and  there  is  left  l  =  k. 


200  OCCULTATIONS  OF  FIXED  STARS  BY  THE  MOON.  223 

And  as  shown  in  Section  XXI.,  a  larger  value  of  this  constant  is , 
required  in  occupations  than  in  eclipses,  to  allow  for  the  increased 
semidiameter  of  the  moon  caused  by  the  lunar  mountains,  irradiation, 
etc.    This  was  apparently  first  discovered  in  the  discussions  of  equa- 
tions of  condition  from  observations  of  occultations. 

The  general  equations  which  CHAUVENET  has  given  in  Art.  341 
are  under  the  form  for  correcting  the  longitude  of  a  place  from  the 
observed  times.  This  is  the  chief  purpose  for  which  the  formulae  in 
this  section  are  generally  used.  They  are  as  follows,  the  notation 
being  the  same  as  for  solar  eclipses,  except  where  otherwise  specified. 

cos  <5  sin  (a  —  a') 

sin  n 
_  sin  (3  —  3')  cos2  %  (a  —  a')  +  sin  (8  +  3')  sin2  &  (a  —  «') 

sin  TT 

Hour  angle  #  =  /*  —  a'  (437) 

A  sin  B  =  p  sin  <pr          1 
A  cos  B  =  p  cos  yf  cos  #  ) 

£  =  p  cos  <pf  sin  #     1 

V  =  A  sin  (B  —  <5')  >  (439) 

C  =  A  cos  (£  —  <?')  J 

If  log  £  is  small,  correct  the  last  three  for  refraction  by  the  Table 
(Section  XXII.). 

m  sin  If  =  x0  — 
m  cos  M=  y0  — 

n  sin  N  =  ; 
n  cos  JV  =  i 

For  jfe,  PETER'S  value  (Art.  182),  k  =  0.272518  ) 
American  Ephemeris,  &  =  0.272506  j 

sin  </>  = (443) 


If  the  local  mean  time  t  was  observed,  take  h  —  3600.00. 
hk  cos  ^       Am  cos  (M —  N) 


(444) 


7i  n 

w  =  T0  —  t  +  T  (445) 

If  the  local  sidereal  time  //  was  observed,  take  h  =  3609.856. 

&  =  P-Q  —  P-  +  T  (446) 

fa  being  the  sidereal  time  at  the  first  meridian. 

The  value  of  k  above  given  is  the  more  recent  determination  of 


224  THEORY   OF   ECLIPSES.  200 

PETERS,  and  is  a  little  smaller  than  OUDEMAN'S,  given  by  CHAUVE- 
NET. 

To  determine  the  longitude  of  a  place,  the  occultation  must  also 
be  observed  at  some  other  place  whose  longitude  is  known,  and  all 
the  observations  must  then  be  combined  to  form  the  equations  of 
condition  ;  for  which  the  reader  is  referred  to  CHAUVENET'S  own 
article,  as  it  is  the  scope  of  the  present  work  to  include  only  such 
computations  as  are  necessary  for  compiling  the  Nautical  Almanac 
and  for  general  prediction.  For  this  reason  the  above  formulae 
would  have  been  omitted,  were  it  not  for  the  fact  that  some  of 
them  are  referred  to  in  previous  pages,  as  well  as  in  the  succeeding 
pages  of  this  section  ;  and,  moreover,  in  order  that  the  connection  of 
these  few  may  be  seen,  the  whole  series  is  given. 

201.  Prediction  for  a  Given  Place.  —  This  is  of  use  in  order  to  be 
prepared  to  observe  an  occultation,  especially  if  by  the  dark  limb  of 
the  moon,  for  either  immersion  or  emersion.  The  time  of  conjunc- 
tion may  be  found  by  the  method  used  for  solar  eclipses  (Art.  21), 
from  the  differences  of  the  quantity  (a  —  a'),  in  which  a!  for  the 
star  is  a  constant  in  formula  (11). 


(447) 


CHAUVEKET'S  formulae,  Art.  345,  are  as  follows : 
The  epoch  hour  Tlf  being  optional,  is  selected  for  the  first  approxi- 
mation at  the  instant  of  conjunction  in  right  ascension,  found  above. 
For  this  time  (a  —  a')  =  0,  whence  equation  (336)  becomes 

x=  0  ^ 

sin  (* —  *')       d  —  d'          .     I  (448) 

y  = ^ J-  —  —      -  nearly,  f 

Sin  7T  7T  ) 

We  also  have  from  the  "  Diff.  for  1  Minute,"  in  the  Nautical  Alma- 
nac, Aa  and  Ad,  the  hourly  motions  are 

xf  =  900  -  —  cos  d  (given  in  arc) 

J"  (449) 

/=    60  - 

7T 

At  the  time  of  contact  the  hour  angle 

#  ==  ^  —  a'  —  a>  (450) 


201  OCCULTATIONS  OF  FIXED  STARS  BY  THE  MOON.  225 

in  which  ^  is  the  sidereal  time  at  the  first  meridian,  a'  the  right 
ascension  of  the  star,  and  a>  the  longitude  of  the  given  place.  We 
also  have 

A  sin  B  =  P  sin  9'  j 

A  cos  B  =  p  cos  (p1  cos  $  ) 

£  =  P  sin  ?'  sin  *     j  ,..„. 

,  =  ^  sin  (5  -  *')  j 
/*'=  54148  sin  1"  log  ji'  =  9.41916  (453) 


>=/,      sn  <5 

m  sin  Jtf  =  x  —  -  £  j          TI  sin  JV=  a;'  —  £'  )  (455} 

m  cosM  =  y  —  ^  j          n  cos  JV=  yr  —  r/  j 

As  now  adopted  by  the  Nautical  Almanac  office  for  occultations, 

&  =  0.272506  log  k  =  9.43538  (456) 


.  f  —  JV)  x.rrrx 

sm  ^  =  -       ^—  —  (457) 

A/ 

k  cos  0      m  cos  (M—  N) 

r  =  ----  (45o) 

n  n 

T=Tl  +  r  (459) 

The  angle  (p  is  obtuse  with  its  cosine  negative  for  immersion,  and 
acute  with  its  cosine  positive  for  emersion,  all  the  quantities  in  the 
first  approximation  being  taken  for  the  time  Tr 

Angles  of  Position.  —  In  solar  eclipses  these  were  measured  on  the 
sun's  limb  toward  the  moon,  but  here  we  wish  the  angle  measured 
on  the  moon's  disk  toward  the  star  ;  it  will  therefore  differ  180°  from 
the  eclipse  value.  Hence,  we  have  the  angles  of  position  from  the 
north  point. 

§=180°-f^+^  (460) 

And  from  the  vertex 


rfO  (461) 

Tf'  ) 


p  COS)'  =  Tfj  -f  Tf) 

Angles  of  position  from  the  vertex  =  180-f-ZV+0  —  r      (462) 

The  distance  of  the  star  from  the  moon's  limb  I  at  the  middle  of 
the  occupation  is  found  thus.  The  distance  of  the  star  from  the 
centre  of  the  moon  is  as  in  former  cases. 


(463) 

in  which  A  is  to  be  taken  as  positive.     J  is  a  fraction  of  the  unit  of 
measure  throughout  this  section,  which  is  n  (see  Section  XXI.); 


226  THEORY   OF   ECLIPSES.  201 

hence  in  seconds  of  arc  it  is  ;rJ.  The  moon's  semidiameter  being  s? 
the  distance  from  the  moon's  limb  is  s  —  x  A  •  but  s  =  for,  therefore 
the  distance  from  the  moon's  limb,  omitting  the  augmentation,  is 

*(*  —  J)*  (464) 

For  a  second  approximation  the  times  resulting  from  the  first 
approximation  become  Tlt  for  which  the  quantities  must  be  again 
taken  from  the  tables,  x  and  y  must  then  be  computed  for  these 
times  from  equation  (436),  or  they  may  be  computed  by  the  following  : 


(465) 
y-jb  +  1(21  -30     ) 

And  &  may  be  found  from  r,  by  reducing  it  from  mean  time  to 
sidereal,  and  then  from  time  to  arc,  and  adding  this  to  the  value  of 
$,  first  found.  CHAUVENET  uses  four-place  logarithms  here,  but 
five-place  would  be  better,  though  the  labor  would  also  be  greater. 

If  in  any  computation  it  is  found  that  sin  <p  >  1,  it  shows  that 
there  is  no  occultation,  for  then  we  have 

A  >  k  (466) 

No  example  of  the  above  formulae  is  here  given,  but  the  reader 
will  find  an  example  given  at  the  end  of  each  Nautical  Almanac,  in 
the  addenda,  "On  the  Use  of  the  Tables,"  the  computation  being 
carried  out  in  full  with  various  precepts  and  explanations. 

202.  The  Limiting  Parallels.  —  These  are  intended  to  show  approxi- 
mately the  portion  of  the  earth's  surface  over  which  an  occultation 
will  be  visible,  by  finding  the  extreme  parallels  of  latitude  between 
which  the  path  across  the  earth  lies.  The  intersection  of  the  cylin- 
der of  rays  from  the  moon,  with  the  fundamental  plane,  we  will  call, 
for  want  of  a  better  word,  the  shadow.  As  in  eclipses,  the  shadow 
moves  across  the  earth  from  west  to  east,  forming  the  shadow  path. 
In  Fig.  28,  Plate  XI.,  the  large  circle  represents  the  earth,  ortho- 
graphically  projected  upon  the  fundamental  plane,  the  shadow  path 
being  that  portion  included  between  the  lines  a  a!  and  b  bf. 

In  occultations,  the  unit  of  measure  for  the  earth's  radius  is  TT,  the 
moon's  parallax  (Section  XXI.)  ;  the  moon  or  shadow  is  k,  a  linear 
fraction  of  ic,  which,  reduced  to  seconds,  becomes  &TT,  the  moon's  semi- 
diameter.  It  is  seen  in  Fig.  28  that  if  the  shadow  falls  so  far  north 

*  CHAUVENET  has  transposed  k  and  A  in  his  text,  so  that  his  equation  will  give 
negative  values,  because  k,  the  radius  of  the  shadow,  must  be  the  greater  of  the  two 
numerically. 

f  Through  some  oversight  in  CHAUVENET'S  text  (Art.  345,  p.  557)  the  term  x0  is. 
omitted  in  these  formula?. 


202  OCCULTATIONS  OF  FIXED  STARS  BY  THE  MOON.  227 

of  the  earth  that  the  southern  line  b  b'  is  tangent  to  the  earth  on  its 
northern  part,  the  distance  of  the  centres  from  one  another  will  be 
TT  +  8  =  61'  28".8  +  16'  44".4  =  1°  18'  13".2 

=  nearly  1°'  20' 

which  expresses  the  approximately  extreme  difference  in  declination 
between  the  moon  and  a  star,  within  which  an  occultation  is  visible. 
This  limit  is  taken  in  round  numbers  at  1°  20r. 

Not  every  star  in  the  heavens  can  be  occulted.  The  inclination 
of  the  moon's  orbit  to  the  ecliptic  is  about  5°  9',  and  when  at  its 
greatest  latitude  north,  a  star  1°  20'  south  of  it  may  cast  a  shadow  at 
the  limit  on  the  north  pole,  while  a  star  1°  20'  north  of  it  will  cast 
a  shadow  at  the  limit  on  the  south  pole,  so  that  the  extreme  distance 
of  a  star  from  the  ecliptic,  in  order  to  be  occulted,  is 

5°  9'  -f  1°  20'  =  6°  29'. 

Only  those  stars  lying  within  this  distance  of  the  ecliptic  can  be 
occulted.  None  others.  As  the  node  of  the  moon's  orbit  has  a 
retrograde  jnotion  on  the  ecliptic,  the  extreme  north  and  south  points 
of  the  orbit  move  slowly  through  this  celestial  belt,  making  a  com- 
plete revolution  during  the  Saros  of  18  years,  or  242  lunations. 

Formerly,  occultations  for  the  Nautical  Almanac  were  computed 
for  even  faint  stars,  but  for  1905  the  limit  of  magnitude  was  fixed 
at  6.5,  and  a  catalogue  of  stars  made  including  all  that  can  possibly 
be  occulted  down  to  this  magnitude.  I  do  not  know,  however,  the 
actual  limits  taken  for  this  star  catalogue. 

In  computing  the  limiting  parallels,  the  first  thing  done  is  to  ascer- 
tain what  stars  will  be  occulted  during  the  year.  With  the  advanced 
pages  of  the  moon's  ephemeris,  as  given  in  the  Almanac,  these  are 
compared  with  the  above-mentioned  catalogue  of  stars,  keeping  the 
right  ascensions  equal,  and  noting  any  star  whose  declination  differs 
from  that  of  the  moon  by  an  amount  not  greater  than  1°  20'.  With 
a  little  practice  this  can  be  done  quite  rapidly,  including  all  doubtful 
stars  for  closer  investigation  afterward,  and  making  a  list  of  their 
names. 

The  limiting  parallels  are  then  computed  by  the  following  for- 
mulae, given  by  CHAUVENET,  Art.  346. 

First  find  the  time  of  conjunction  by  equation  (447),  for  which 
the  moon's  right  ascension  for  four  hours  must  be  taken  from  the 
Nautical  Almanac  pages,  for  which  time  the  quantities  in  the  for- 
mulae must  be  interpolated. 

<5  — <?' 
2A>= (468) 


228  THEORY   OF   ECLIPSES.  202 

x>  =  900  —  cos  d 

(469) 

60  " 


°  (470) 

j 
And  the  limiting  parallels  <pl  and  <p2  are 

cos  YI  =  2/o  sin  JV±  0.2723  (n  <  180°)  ) 

sin  /?  =  sin  N  cos  <5'  (/J  <  90°)    V  (471 ) 

?i  =  /?±n  J 

cos  ft  =  2/0  sin  ^V=F  0.2723  ) 

sin  ?t  =  sin  (JV  =F  ft)  cos  *'  JV  <  90°  j 

If  the  star's  declination  is  north,  the  upper  signs  are  to  be  used 
throughout,  and  ^  is  the  northern  and  <p2  the  southern  parallel.  If 
the  declination  is  south,  use  the  lower  signs,  and  <pl  is  the  southern 
and  <p2  the  northern  parallel. 

N  is  to  be  considered  as  positive  and  less  than  90°,  although  the 
angle  may  strictly  be  obtuse,  the  reason  for  which  will  be  seen  below. 

To  the  above  are  also  appended  the  following  precepts  :  When  all 
the  shadow  does  not  fall  upon  the  earth,  as  shown  by  cos  Y\  °r  cos  f2 
being  imaginary. 

(1)  Cos   YI  Imaginary. — The    occultation   is   visible   beyond   the 
elevated  pole  (north  or  south),  and  +  90  in  north  latitude  or  —  90 
in  south  latitude  is  the  conventional  designation  that  the  occultation 
is  visible  beyond  the  pole  for  a  distance  in  degrees  equal  numerically 
to  the  star's  declination. 

(2)  Cos  f2  Imaginary. — The  occultation  is  visible  in  the  vicinity 
of  the  depressed  pole  (north  or  south).     If  the  star's  declination  is 
north,  the  south  pole  is  depressed  and  the  limiting  parallel  is  <p2  — 
d'  —  90°.     If  the  declination  is  south,  the  north  pole  is  depressed 
and  the  limiting  parallel  is  <f>2  =  8'  +  90°.     The  algebraic  signs  of 
dr  must  here  be  regarded. 

(3)  Ifyl=p±rl  exceeds  90°,  the  true  value  is  either  ^  —  180  — 
(/3  db  ft)  or  ^  =  —  180°  —  (/?  =h  f),  since  these  have  the  same  sine. 

203.  Example. — We  will  take  the  star  46  Yirginis  and  the  date 
1905,  Sept.  1.  This  example  includes  a  little  more  than  the  limiting 
parallels,  for  it  was  computed  for  the  Nautical  Almanac,  and  gives 
all  the  elements  of  the  occultation  as  usually  published. 

First  take  from  the  moon's  ephemeris  the  right  ascension  for  four 
hours,  two  before  and  two  after  the  approximate  time  of  conjunction; 


203  OCCULTATIONS  OF  FIXED  STARS  BY  THE  MOON.  229 

subtracting  from  these  the  star's  reduced  right  ascension  gives  a.  —  #', 
with  which  the  correct  Greenwich  mean  time  of  conjunction  is  to  be 
gotten  by  formula  (447),  which  time  is  placed  at  the  head  of  the 
example.  The  Washington  mean  time  is  found  by  deducting  from  it 
the  longitude  of  Washington  in  time.  This  is  then  reduced  to  sidereal 
time  (s.  t.).  The  hour  angle  &  is  then  found  by  (473)  ;  the  sidereal 
time  at  Greenwich  noon  is  corrected  for  the  sidereal  interval  cor- 
responding to  the  longitude  5A  8W.26  from  Table  III.,  at  the  end  of 
the  Nautical  Almanac,  giving  the  sidereal  time  at  Washington  mean 
noon.  Adding  to  this  the  above  Sh  26m.2  gives  the  Washington  mean 
time  of  conjunction. 

OCCULTATION   OF   46   VlRGINIS,  1905,    SEPT.  1. 

LIMITING  PARALLELS  AND  DATA  FOR  THE  Nautical  Almanac. 

[References  to  pages  are  to  the  Nautical  Almanac,  1905.] 


Elements  for  Nautical  Almanac. 

For  Limiting  Parallels. 

Star's  name,  Magnitude,  6.1,  46  Virginis. 

(468)  log  (d-d')    3002".8  +3.47752 

log  TT                                 3.56003 

(447) 

CT  G.  M.  T.                             13*  33-1 
w    Washington,  p.  524              5     8  .26 

N.  A.  y0                  +0.8270  +9.91749 

N.A. 

O'  W*.  M.  T.  (N.  A.,  p.  470)      8  24  .8 

(469)  Diff.  for  1«      28.3390      0.36903 

Reduc.  to  s.  t.,  p.  592              +1  .4 

cos  6                                9.99973 

C/  in  s.  t.                                    8   26  .2 

15  X  60  =  900                2.95424 

In  arc.                               3.32300 

s.  t.  at  G.  M.  N.,  p.  147      10*  39^.92 

TT                                        3.56003 

Reduc.  for  5*  8.26,  p.  592           0  .85 

N.  A.  z'                      0.5794      9.76297 

s.  t.  W.  M.  N.                     10  40  .8 

W.M.T.ofcf                   19     7  .0 

Diff  for  1»  —  12".027  —1.08016 

60                         1.77815 

N.A. 

^-  R.  A.  (mean  place)     12*  55"*  42S.357 

1  :  TT                                  6.43997 

N.A. 

Aa                                                   +0  .560 
^  Reduced  R.  A.,  a'      12   55    42.92 

N.  A.  y'                  —0.1987  —9.29828 

N.A. 
N.A. 

-^  Dec.  (mean  place)      —2°  51'  27".78 
M                                               —0  .94 

(470)  tan  N                              0.46469 
N  (acute)                     +71°  4' 

N.A. 

%•  Reduced  Dec.  d'         —2   51   28  .72 

(471)  sin  JV                           +9.97584 

(473) 

^  R.  A.  as  above.              a    12*  55TO.7 

2/0  sin  -ZV        +0.7822  +9.89333 
k                    —0.2723 

N.A. 

Hour  angle.     H=tf  —   a  +6   11  .3 

cos  7l             +0.5099 

3  Dec.  at  13*                  —1°  54'  47"  7 

7i                  _|_5go  20' 

Interpol.  for  tf                 —6   38  .3 
6                                 —2      1   26  .0 

sin  lY                                9.97584 

6  —  <?'                        +0    50     2  .7 

sin  N  cos  <J'  =  sin  J3        9.97557 

/?                    +70  58 

$1  =  13  —  7i   +11  38 

(472)  cos  y2               1.0545  Imaginary. 

Case  2. 

0a  ==  d'  +  90°  =  —  2°  51'  +  90° 

—  _|_  87°  9' 

230  THEORY  OF   ECLIPSES.  203 

The  star's  mean  right  ascension  and  declination  (which  are  placed 

in  the  Almanac  on  page  445)  are  then  reduced  to  the  given  date  for 

proper  motion,  annual  variation,  precession,  etc.     With  the  star's 

right  ascension  first  find  the  hour  angle  as  gotten  for  the  Almanac. 

&or  H=cf  W.M.N—af  (473) 

The  moon's  declination  is  interpolated  for  the  time  of  conjunction 
and  d  —  d'  then  gotten.  This  completes  the  first  column  of  the 
example,  which  gives  chiefly  the  data  to  be  inserted  in  the  Almanac 
(pages  445  and  470),  and  also  the  data  to  be  used  in  the  formulae  for 
the  limiting  parallels. 

Formulae  (468-72)  are  now  to  be  computed  for  the  limiting  paral- 
lels. It  is  first  to  be  noted  that  Aa  and  Ad  in  the  first  column  of  the 
example  are  the  Nautical  Almanac  notation  for  the  small  reductions 
of  the  star's  place,  and  are  given  on  page  470  of  the  Almanac  under 
that  notation.  But  Aa  and  Ad  in  the  second  column  of  the  example 
and  in  formulae  (467)  are  quite  different  quantities,  being  the  "  diff.  for 
1  minute  "  of  the  moon's  motion  given  in  the  Nautical  Almanac  on 
page  150.  Nj  the  angle  which  the  path  makes  with  the  axis  of  Yy 
is  to  be  taken  less  than  90°,  though  the  formulae  may  strictly  give  it 
obtuse,  as  they  do  in  this  example.  The  constants  900  and  60  arc 
15,  to  reduce  the  right  ascensions  from  time  to  arc,  and  60  to  reduce 
Aa  and  Ad  from  the  motion  in  one  minute  to  the  motion  in  one  hour, 
so  that  x'  and  y'  can  be  given  in  the  Almanac  as  hourly  motions. 
Five-place  logarithms  are  used  here,  and  y0,  #',  yr  are  given  to  four 
places  of  natural  numbers  for  the  Almanac. 

In  formulae  (471)  and  (472),  the  star's  declination  being  south,  the 
lower  signs  are  to  be  used,  and  <pl  is  the  southern  parallel  +11°.  ft 
may  be  obtuse,  but  in  the  example  it  is  acute. 

f2  results  >  unity,  which  comes  under  Case  2.  The  declination 
being  south,  the  north  pole  is  depressed,  the  shadow  extends  beyond 
the  earth,  and  the  parallel  furthest  from  the  equator,  which  is  above 
(or  in)  the  fundamental  plane,  is  d  -f  90°  =  +  87°. 

For  this  example,  besides  various  other  suggestions,  I  am  indebted 
to  Mr.  JAMES  ROBERTSON,  of  the  Nautical  Almanac  office,  who  now 
computes  the  Occultations,  and  who  kindly  placed  his  computing 
sheets  at  my  disposal.  Mr.  ROBERTSON  has  transformed  CHAUVE- 
NET'S  formulae  and  devised  a  shorter  method  for  these  limiting  paral- 
lels, with  the  use  of  manuscript  tables ;  and  on  that  account  I  have 
been  obliged  to  deviate  from  the  latter  part  of  his  work  and  keep 
to  CHATJVENET'S  formulae.  It  is  hoped  that  he  will  publish  his 
method  and  tables. 


204  OCCULTATIONS  OF  FIXED  STARS  BY  THE  MOON.  231 

204.  Graphic  Representation. — The  declination  in  the  above  exam- 
ple is  so  small,  all  parts  of  the  construction  following  could  not  well 
be  seen,  so  that  for  illustration  I  have  selected  another  occupation 
from  the  Nautical  Almanac,  1905,  page  450,  Feb.  10. 

85  Ceti  Mag.  6.3  cf  21*  29W.3  3'  =  +  10°  20'.1 

2/o  =  +  0.2043  x'  =  -f  0.5297  y'  =  +  0.1461 

Limiting  parallels,         +  47°  and  —  19° 

In  Fig.  28,  Plate  XI.,  let  the  larger  circle  represent  the  orthographic 
projection  of  the  earth  upon  the  fundamental  plane.  The  radius  of  the 
circle  is  the  unit  of  measure,  and  in  arc  is  the  moon's  parallax  it.  Z  is 
the  zenith  of  the  projection,  as  in  previous  figures.  The  shadow  passes 
from  left  to  right  over  the  earth,  and  at  the  time  of  conjunction  is  at 
the  point  K  on  the  axis  of  Y.  ZK  therefore  represents  d  —  3f  in 
arc  proportionate  to  the  radius  of  the  earth's  sphere,  or  dividing  by 
'  TT,  it  is  a  decimal  fraction,  the  quantity  y0  =  -f  0.2043  in  formula 
(468).  If  we  lay  off  ZA  =  x'  =--  +  0.5299,  the  motion  of  the  shadow 
in  right  ascension  in  one  hour,  and  erect  AD  =  y'  =  -f-  0.1461 
(equation  (469)),  the  motion  in  declination  in  one  hour,  ZD  will 
represent  the  direction  of  the  shadow  path.  Drawing  a  line  through 
K  parallel  to  this  last,  we  have  MR,  the  path  of  the  shadow  across 
the  earth.  If  two  other  lines  aaf  and  bbf  be  drawn  parallel  to  MR 
and  distant  from  it  =  k  =  0.2723,  they  will  be  the  limiting  lines  of 
the  shadow  path.  Let  P  be  the  north  pole,  elevated  in  this  figure, 
which  shows  that  the  star's  declination  is  north,  then  PKR  —  the 
angle  N  (equation  (470))  ;  also,  PZQ  =  N. 

If  the  plane  of  YZ,  which  is  perpendicular  to  the  plane  of  the 
paper,  be  revolved  about  the  axis  of  Y  through  an  angle  of  90°  to 
the  right,  Z  will  fall  at  F9  NS  will  represent  arc  Na  bs  in  the  funda- 
mental plane,  and  the  pole  P  will  fall  at  some  point  Pr.  We  can 
therefore  lay  off  any  desired  declination.  NZP'  =  d'  =  -f  10°  20', 
and  when  revolved  back,  Pf  will  fall  at  P,  whose  position  we  now 
know.  It  is  seen  that  as  ZN  or  ZP1  is  taken  as  unity,  ZP  will  be 
the  cosine  of  the  declination. 

Drawing  ZH  parallel  to  the  path  and  also  ZI  and  PQ  perpen- 
dicular to  the  path,  we  have 

ZI  =  ZK  sin  N=  yQ  sin  N 

Formulae  (471)  and  (472)  should  be  closely  followed  in  connection 
with  these  pages.  We  have  also  the  following  quantity,  which  is 
arbitrarily  called  cos  ft. 


232  THEORY   OF   ECLIPSES.  204 

ZC  =  cos  ft  =  2/o  sin  N  +  0.2723         (Equa.  (471)) 
We  use  the  -j~  sign  because  the  declination  is  north. 

Also,  PQ  =  PZ  sin  N 

or  sin  /?  =  sin  N  cos  df  (Equa.  (471)) 

The  angles  ft  and  /9  will  now  be  shown  and  their  sum.  Draw  Za, 
which  is  the  radius  of  the  sphere ;  CZ,  being  the  cosine  of  ft,  CZa 
must  be  the  angle  ft  itself.  Also,  since  PQ  or  its  equivalent  02T  = 
sin  /9,  draw  06r,  parallel  to  the  path,  passing  through  P  of  course, 
and  meeting  the  circle  in  Gr,  then  GZH=ft.  This  angle  must  be 
revolved  round  the  point  Z  so  that  G  will  fall  in  C' ;  therefore, 
laying  off  the  arc  HT  =  GCf,  we  have  /?  =C'ZT;  and  ft,  now  lay- 
ing adjacent  thereto,  aZT=^-\-  ft,  an  obtuse  angle  in  which  by  Case 
III.  <pv  is  its  supplement,  which  exactly  equals  47°,  given  in  the 
Nautical  Almanac;  its  sine  is  the  perpendicular  dropped  from  T 
upon  aZ  produced. 

The  southern  parallel  is  found  thus : 

cos  ft  =  2/0  sin  N  —  0.2723  =  ZE, 

a  negative  quantity  in  this  example  which  gives  the  obtuse  angle 
C'Zb.  It  is  also  seen  from  the  figure  that  C'ZL  is  the  angle  N9 
because  this  angle  and  also  NZH  are  both  equal  to  90°  minus  the 
arc  C'N.  Hence, 

LZb  =  C'ZL  —  C'Zb  =  N—r* 

This  arc  Lb  is  the  hypothenuse  of  a  spherical  triangle  whose  base 
lies  in  the  meridian  passing  through  the  equinoxial  point  Z/,  and 
which  is  therefore  the  latitude  of  the  point  b.  Likewise  the  sine 
of  (N —  f2)  is  the  hypothenuse  of  a  plane  triangle  of  which  the  base 
is  sin  (N — ft)  cos  df.  This  shows  the  meaning  of  this  factor  in  these 
formulae.  We  can  lay  off  the  angle  d'  from  the  point  L,  and  a  per- 
pendicular from  b  let  fall  upon  it  gives  the  sine  of  the  latitude.  In 
the  example  this  point  falls  between  the  two  lines  shown  in  the 
figure. 

It  is  plainly  evident  that  the  formulae  give  the  southern  limiting 
parallel,  because  the  point  lies  in  the  fundamental  plane  and  the  con- 
struction is  more  easy  in  consequence.  It  is  not  so  plainly  seen  that 
the  formulae  give  the  northern  parallel,  because  the  point  is  in  space 
and  the  construction  more  difficult  to  show  the  latitude  itself. 

In  Fig.  28  we  can  see  some  of  the  other  conditions  attached  to  the 
formulae  when  the  whole  shadow  does  not  fall  upon  the  earth.  At 


204  OCCULTATIONS  OF  FIXED  STAKS  BY  THE  MOON.  233 

the  north  pole,  the  pole  being  elevated  in  the  figure,  an  occupation 
is  seen  over  and  beyond  the  pole,  and  in  fact  all  around  it  as  far  as 
the  parallel  90  —  df,  so  that  90°  is  the  conventional  northern  limit 
here. 

If  the  shadow  should  go  off  the  earth  at  the  south  pole,  which  is 
depressed,  the  nearest  parallel  to  the  pole  is  numerically  90  —  d,  but 
the  formula  d  —  90  gives  the  parallel  algebraically  whether  north  or 
south.  The  shadow  cannot  reach  further  south  than  this,  for  that 
portion  of  the  earth  is  below  the  horizon.  The  condition  for  the 
shadow  falling  partly  off  the  earth  is  ft  or  f2  greater  than  unity ;  this 
gives  the  limiting  lines  of  the  path,  as  above  shown,  falling  beyond 
the  earth,  whose  radius  is  unity. 

In  the  elements  of  occultation,  therefore,  when  we  see  3f  and  yf 
(the  motion  north  or  south)  having  the  same  sign,  we  may  expect  to 
see  90°  for  some  of  the  limiting  parallels ;  but  when  df  and  yf  have 
different  signs,  the  shadow  may  go  off  the  earth  at  the  depressed 
pole,  which  will  be  known  by  examining  d'  for  the  condition,  <p  = 
90  —  df,  or  as  we  have  in  the  Nautical  Almanac,  <p  and  d,  <p  -f-  d  = 
90°.  If  the  shadow  partly  falls  off  the  earth,  yQ  is  generally  greater 
than  unity ;  but  if  the  path  is  much  inclined  (yr  large),  the  limit  is 
greater. 

An  occultation  if  shown  in  a  chart  like  the  eclipses,  would  be 
very  similar  to  an  eclipse,  the  rising  and  setting  curve,  northern  and 
southern  limits,  outline,  etc.,  as  shown  in  Figs.  17  and  18,  Plates 
VI.  and  VII.,  but  there  would  be  no  central  line  or  curves. 

205.  It  is  possible  to  find  the  limiting  parallels  graphically  by  the 
method  of  descriptive  geometry.  Fig.  29,  Plate  XI.,  shows  the 
earth's  sphere  and  path  of  the  shadow,  as  in  the  previous  figure. 
Consider  a  vertical  plane  passed  through  the  line  aaf,  the  northern 
limit  of  which  we  want  to  find  the  limiting  parallel.  Revolve  this 
plane  toward  the  lower  part  of  the  drawing.  The  plane  will  cut 
from  the  earth  a  small  circle,  which  is  seen  passing  diametrically 
through  the  points  a  and  af.  We  now  want  the  position  of  the 
north  pole.  In  the  revolution  it  will  fall  in  the  indefinite  line  PC, 
drawn  perpendicular  to  the  axis  of  revolution  aaf  and  crossing  it  at 
D.  In  the  previous  figure  we  find  the  height  of  the  pole  above  the 
fundamental  plane  to  be  PPf.  Drawing  the  line  PE  =  PPf  perpen- 
dicular to  the  line  PC,  the  distance  of  the  pole  from  the  axis  aar, 
which  is  a  line  in  space  before  the  revolution,  is  DE.  An  observer 
at  a  would  see  the  pole  revolve  from  E  through  an  arc  of  90°  to  F, 


234  THEORY   OF   ECLIPSES.  205 

and  the  projection  of  its  path  on  the  plane  of  the  paper  is  PC,  C 
being  found  by  the  perpendicular  FG,  so  that  in  the  revolution  of 
this  portion  of  the  sphere  the  small  circle  represents  every  portion 
of  the  earth  over  which  the  northern  limiting  line  has  passed.  And 
C  being  the  north  pole,  it  is  seen  that  the  most  northerly  parallel  of 
latitude  touched  by  the  northern  limit  of  the  path  is  that  portion  of 
this  small  circle  which  is  nearest  to  the  pole.  Drawing  from  the  centre 
of  this  circle,  J,  the  line  JC,  it  meets  this  circle  at  H  and  Hr,  and 
the  polar  distance  of  the  limiting  parallel  is  the  arc  CH.  This  we 
must  now  find. 

We  will  again  revolve  this  section  of  the  sphere  about  the  line 
HHr  (which  is  in  the  fundamental  plane),  these  being  the  points  in 
which  this  axis  intersects  the  small  circle.  The  highest  point  of  the 
surface,  which  is  over  the  centre,  J,  will  fall  in  the  line  JI  drawn 
perpendicular  to  HH',  at  a  distance  equal  to  that  shown  in  Fig.  28, 
CC'j  since  HHf  and  CC'  in  both  figures  are  similar  cords  of  the 
circle,  and  both  lie  in  the  fundamental  plane.  The  centre  of  the 
sphere  will  fall  in  the  line  JI  produced  backward.  The  north  pole 
falls  also  in  a  line  perpendicular  to  the  axis  of  revolution  at  a  dis- 
tance, CL  =  FC.  The  points  H  and  H'  remain  fixed.  The  sphere 
can  be  drawn  from  the  centre  above  found,  and  should  pass  through 
these  four  points,  jff,  H',  I,  and  L,  the  pole. 

The  sphere  as  now  represented  shows  the  north  pole,  Z,  being  in 
the  fundamental  plane,  the  sphere  projected  upon  a  meridian  passing 
through  the  point  of  tangency,  H,  of  the  path  and  limiting  parallel, 
whose  polar  distance  is  consequently  the  arc  LH. 

The  southern  parallel  is  very  easily  shown,  since  it  lies  in  the 
fundamental  plane.  When  the  sphere  was  revolved  as  above  and 
the  pole  P  fell  at  P',  the  point  b  of  the  southern  parallel  fell  at  bn '. 
Draw  through  this  point  the  line  cc',  and  through  the  point  Z  the  line 
ee'9  both  perpendicular  to  P'Z;  the  former  will  be  the  parallel  of 
latitude  and  the  latter  the  equator  in  their  revolved  position,  and  the 
latitude  e  c  or  efc'  can  be  measured  by  scale. 

The  occultations  have  generally  taken  up  the  entire  time  of  one 
person  throughout  the  year  in  the  Nautical  Almanac  office,  and  some- 
times two  or  three  have  been  engaged  upon  the  computations.  The 
author  has  paid  comparatively  little  attention  to  this  division  of  the 
subject  treated  of  in  this  work,  his  time  being  fully  occupied  with 
the  eclipses  and  the  computations  for  the  planets  Mercury,  Venus, 
Mars,  Jupiter,  Saturn,  Uranus,,  and  Neptune. 


206  OCCULTATIONB  OF  FIXED  STARS  BY  THE  MOON.  235 

206.  Criterion  of  visibility  at  any  place,  as  given  in  the  Almanac 
(see  page  576,  for  1905). 

(1)  The  limiting  parallels  must  include  the  latitude  of  the  place. 

(2)  The  quantity  H —  A  (in  the  Almanac  notation,  or  H —  co  of 
the  present  work),  taken  without  regard  to  sign,  must  be  less  than 
the  semidiurnal  arc  of  the  stars  by  at  least  one  hour.     Upon  very 
rare  occasions  an  emersion  might  be  seen  in  the  east,  or  an  immer- 
sion in  the  west,  when  this  difference  is  a  few  minutes  less  than  an 
hour. 

(3)  The  sun  must  not  be  much  more  than  an  hour  above  the  hori- 
zon at  the  local  mean  time,  T —  X  [T —  ai],  unless  the  star  is  bright 
enough  to  be  seen  in  the  daytime. 

Fig.  28  shows  how  unsatisfactory  are  the  limiting  parallels  in 
determining  the  locations  on  the  earth  where  the  occultation  will 
be  visible ;  in  the  space  shown  in  the  figure  above  the  northern 
limit  and  the  much  larger  space  below  the  southern  limit,  the 
occultation  is  not  visible  at  all.  The  visibility  does  not  include 
more  than  one-third  or  possibly  one-half  of  the  earth's  surface 
contained  between  the  limiting  parallels. 

Several  graphic  methods  have  been  devised  and  published  for 
narrowing  the  zone  of  visibility,  so  as  to  avoid  the  tedious  compu- 
tation. The  orthographic  method  described  in  Section  XIX.,  the 
method  by  semidiameters,  which  is  used  here  in  Fig.  28,  is  sug- 
gested. This  could  be  drawn  on  a  much  larger  scale,  and  the  axes 
marked  into  tenths  or  smaller  divisions  of  the  radius,  especially  the 
axis  of  X  between  the  values  0.515  and  0.600,  which  are  the  extreme 
values  of  the  coordinate  xr ;  and  Y  from  0  to  ±0.200,  the  extreme 
values  yf.  These  should  be  marked  in  both  directions,  and  the 
divisions  for  y'  carried  beyond  the  sphere  to  ±1.40  for  yoy  which 
value  has  nearly  been  reached  by  some  occupations. 

The  direction  of  the  path  can  be  shown  by  a  thread  laid  over  the 
proper  values  of  x',  y',  taken  from  the  Almanac,  and  a  line  parallel 
thereto  drawn  through  the  proper  value  of  yQ  gives  the  path,  which 
can  be  done  without  measuring.  The  limiting  lines  are  distant  from 
the  path  0.2723,  a  constant. 

The  ellipse  of  the  given  place  then  is  needed  with  hour  angles 
marked  on  it.  The  location  of  the  given  place  can  be  determined 
by  its  hour  angle.  The  Washington  hour  angle  and  the  time  of 
conjunction  are  both  given  in  the  Almanac.  The  method  is  precisely 
that  described  in  the  section  on  Method  by  Semidiameters,  Section 
XIX.,  which  is  illustrated  in  Fig.  21,  Plate  VIII. 


236  THEORY   OF   ECLIPSES.  206 

The  following  is  also  suggested  as  a  substitute  for  the  limiting 
parallels :  That  instead  of  these,  the  Nautical  Almanac  might  give 
the  latitudes  and  longitudes  of  three  or  four  points  on  each  of  the 
two  limiting  lines  of  the  path,  which,  when  plotted  on  a  map,  would 
show  much  more  closely  and  clearly  the  regions  in  which  the  occulta- 
tion  is  visible. 


LIST  OF  TABLES 

AND  KEFERENCES  TO  PAGES. 


TABLE 

I.  Reduction  of  time  to  arc 34 

II.  Interpolation  for  the  eclipse  hours 26,  34 

III.  Interpolation,  Coefficient  for  second  differences 26,  34 

IV.  Logarithm  of  the  Earth's  radius 68,  73,  158 

V.     Table  giving  log —^— 43 

1  — 6 

VI.     Sidereal  interval  for  the  mean  time  hours 43 

VII.    Giving  log /*/ 43-4 

VIII.     Proportional  parts  for  ^ 67-8 

IX.     Sine  and  Cosine  for  every  5° 95 

X.     Giving  e.     Formula  for  Outline 94-5 

XI.     Quantities  used  in  HILL'S  Formulae 101 

XII.  Reduction  to  Geocentric  Latitude  and  Angle  of  the  Vertical    .    .    .  145,  158 

XIII.  Northern  and  Southern  Limiting  Curves,  For  v' 108-9-10 

XIV.  Part  I.  and  II.,  Northern  and  Southern  Limiting  Curves,  For  Q  .     108-9-10 
XV.  The  total  number  of  eclipses  in  one  Saros,  1889  to  1906,  classified  by 

years 64-5 

XVI.  List  of  all  Total  and  Annular  Eclipses  in  one  Saros,  with  precepts  for 

finding  their  approximate  locations  for  future  years 61 

ADVERTISEMENT  ;  On  the  Projection  of  the  Sphere 143-160 

237 


238 


TABLES. 


Table  I. — KEDTJCTION  OF  TIME  TO  AKC. 


Hours. 

Minutes. 

Seconds. 

Decimals. 

0*     0° 

0™0°    0' 

30m    70  30/ 

0s  0'    Ox/ 

30s    7'  30" 

O'.OOCX'.OO  08.50   7".50 

1     15 

1    0    15 

31      7    45 

1 

0  15 

31     7  45 

.01  0  .15 

.51    7  .65 

2    30 

2    0    30 

32      8      0 

2 

0  30 

32     8    0 

.020  .30 

.52    7  .80 

3    45 

3    0    45 

33      8    15 

3 

0  45 

33     8  15 

.030  .45 

.53    7  .95 

4    60 

410 

34      8    30 

4 

1     0 

34     8  30 

.040  .60 

.54    8  .10 

.050  .75 

.55    8  .25 

5    75 

5    1    15 

35      8    45 

5 

1   15 

35     8  45 

.06  0  .90 

.56    8  .40 

6    90 

6    1    30 

36      9      0 

6 

1  30 

36     9     0 

.07  1  .05 

.57    8  .55 

7  105 

7    1    45 

37      9    15 

7 

1  45 

37     9  15 

.081  .20 

.58    8  .70 

8  120 

8    2    00 

38      9    30 

8 

2     0 

38     9  30 

.091  .35 

.59    8  .85 

9  135 

9    2    15 

39      9    45 

9 

2  15 

39     9  45 

.101  .50 

.60    9  .00 

10  150 

10    2    30 

40    10      0 

10 

2  30 

40  10     0 

.11  1  .65 

.61    9  .15 

11  165 

11    2    45 

41    10    15 

11 

2  45 

41   10  15 

.121  .80 

.62   9  .30 

12  180 

12    3      0 

42    10    30 

12 

3     0 

42  10  30 

.131  .95 

.63    9  .45 

13  195 

13    3    15 

43    10    45 

13 

3  15 

43  10  45 

.142  .10 

.64   9  .60 

14  210 

14    3    30 

44    11      0 

14 

3  30 

44  11     0 

.152  .25 

.65    9  .75 

.162  .40 

.66   9  .90 

15  225 

15    3    45 

45    11    15 

15 

3  45 

45  11   15 

.172  .55 

.67  10  .05 

16  240 

16    4     0 

46    11    30 

16 

4     0 

46  11  30 

.182  .70 

.68  10  .20 

17  255 

17    4    15 

47    11    45 

17 

4  15 

47   11  45 

.19  2  .85 

.69  10  .35 

18  270 

18    4    30 

48    12      0 

18 

4  30 

48  12     0 

19  285 

19    4    45 

49    12    15 

19 

4  45 

49  12  15 

.203  .00 

.7010  .50 

.21  3  .15 

.71  10  .65 

20  300 

20    5      0 

50    12    30 

20 

5     0 

50  12  30 

.223  .30 

.7210  .80 

21  315 

21    5    15 

51    12    45 

21 

5  15 

51  12  45 

.23  3  .45 

.7310  .95 

22  330 

22    5    30 

52    13      0 

22 

5  30 

52  13     0 

.24  3  .60 

.7411  .10 

23  345 

23    5    45 

53    13    15 

23 

5  45 

53  13  15 

.253  .75 

.7511  .25 

24  360 

24    6      0 

54    13    30 

24 

6     0 

54  13  30 

.263  .90 

.7611  .40 

.274  .05 

.7711  .55 

25    6    15 

55    13    45 

25 

6  15 

55  13  45 

.284  .20 

.7811  .70 

26    6    30 

56    14     0 

26 

6  30 

56  14     0 

.294  .35 

.7911  .85 

27    6    45 

57    14    15 

27 

6  45 

57  14  15 

28    7      0 

58    14    30 

28 

7     0 

58  14  30 

:304  .50 

.80  12  .00 

29    7    15 

59    14    45 

29 

7  15 

59  14  45 

.31  4  .65 

.8112  .15 

.324  .80 

.82  12  .30 

30    7    30 

60    15      0 

30 

7  30 

60  15     0 

.334  .95 

.8312  .45 

Keduce  11*  12m  24S.983  to  arc. 

08.000        0".000 

.345  .10 
.35  5  .25 

.84  12  .60 
.85  12  .75 

11*                     165° 

1        0  .015 

.365  .40 

.8612  .90 

12™                  3     0 

2        0  .030 

.37  5  .55 

.87  13  .05 

24*                    6        0 

3        0  .045 

.385  .70 

.88  13  .20 

.98                       14.70 

4        0  .060 

.395  .85 

.89  13  .35 

.003                         .045 

5        0  .075 

6                 ft       f\Qf\ 

11*  12™  24'.983  =  168°  6'  14".745 

u  .uyu 
7        0  .105 

.406  .00 
.41  6  .15 

.90  13  .50 
.91  13  .65 

8        0  .120 

.426  .30 

.9213  .80 

9        0  .135 

.436  .45 

.93  13  .95 

0.010        0  .150 

.44  6  .60 

.9414  .10 

.45  6  .75 

.9514  .25 

.466  .90 

.9614  .40 

.477  .05 

.97  14  .55 

.487  .20 

.98  14  .70 

.49  7  .35 

.99  14  .85 

.507  .50 

1  .00  15  .00 

TABLES. 


239 


Table  II. — FOR  INTERPOLATING  THE  SUN'S  RIGHT  ASCENSION 
AND  DECLINATION,  LOG  rr  AND  MOON'S  PARALLAX  FOR 
THE  ECLIPSE  HOURS. 


A2 

1 

23 

2 
22 

3 

21 

4 
20 

5 

19 

6 

18 

7 
17 

8 
16 

9 
15 

10 
14 

11 

13 

12 

O'.OO 

.0000  .0000 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

.05 

.0010  .0019 

.0027 

.0035 

.0041 

.0047 

.0052 

.0055 

.0059 

.0061 

.0062 

.0062 

.10 

.0020 

.0038 

.0055 

.0069 

.0082 

.0094 

.0103 

.0111 

.0117 

.0122 

.0124 

.0125 

.15 

.0030 

.0057 

.0082 

.0104 

.0124 

.0141 

.0155 

.0167 

.0176 

.0182 

.0186 

.0187 

.20 

.0040 

.0076 

.0109 

.0139 

.0165 

.0187 

.0207 

.0222 

.0234 

.0243 

.0248 

.0250 

.25 

.0050 

.0095 

.0137 

.0173 

.0206 

.0234 

.0258 

.0278 

.0293 

.0304 

.0310 

.0312 

.30 

.0060 

.0115 

.0164 

.0208 

.0247 

.0281 

.0310 

.0333 

.0352 

.0365 

.0372 

.0375 

.35 

.0070 

.0134 

.0191 

.0243 

.0288 

.0328 

.0362 

.0389 

.0410 

.0425 

.0434 

.0437 

.40 

.0080 

.0153 

.0219 

.0278 

.0330 

.0375 

.0413 

.0444 

.0469 

.0486 

.0497 

.0500 

.45 

.0090 

.0172 

.0246 

.0312 

.0371 

.0422 

.0465 

.0500 

.0527 

.0547 

.0559 

.0562 

.50 

.0100 

.0191 

.0273 

.0347 

.0412 

.0469 

.0516 

.0555 

.0586 

.0608 

.0621 

.0625 

.55 

.0110 

.0210 

.0301 

.0382 

.0453 

.0516 

.0568 

.0611 

.0645 

.0668  .0683 

.0687 

.60 

.0120 

.0229 

.0328 

.0416 

.0494 

.0562 

.0620 

.0667 

.0703 

.0729  .0745 

.0750 

.65 

.0130 

.0248 

.0356 

.0451 

.0536 

.0609 

.0671 

.0722 

.0762 

.0790 

.0807 

.0812 

.70 

.0140 

.0267 

.0383 

.0486 

.0577 

.0656 

.0723 

.0778 

.0820 

.0851 

.0869 

.0875 

.75 

.0150 

.0280 

.0410 

.0520 

.0618 

.0703 

.0775 

.0833 

.0879 

.0911 

.0931 

.0937 

.80 

.0160 

.0306 

.0438 

.0555 

.0659 

.0750 

.0826 

.0889 

.0938 

.0972 

.0993 

.1000 

.85 

.0170 

.0325 

.0465 

.0590 

.0700 

.0797 

.0878 

.0944 

.0996 

.1033 

.1055 

.1062 

.90 

.0180 

.0344 

.0492 

.0625 

.0742 

.0844 

.0930 

.1000 

.1055 

.1094 

.1117 

.1125 

1  .00 

0.020 

0.038 

0.055 

0.069 

0.082 

0.094 

0.103 

0.111 

0.117 

0.122 

0.124 

0.125 

2.00 

0.040 

0.076 

0.109 

0.139 

0.165 

0.187 

0.207 

0.222 

0.234 

0.243 

0.248 

0.250 

3.00 

0.060 

0.115 

0.164 

0.208 

0.247 

0.281 

0.310 

0.333 

0.352 

0.365 

0.372 

0.375 

4.00 

0.080 

0.153 

0.219 

0.278 

0.330 

0.375 

0.413 

0.444 

0.469 

0.486 

0.497 

0.500 

5.00 

0.100 

0.191 

0.273 

0.347 

0.412 

0.469 

0.516 

0.556 

0.586 

0.608 

0.621 

0.625 

6.00 

0.120 

0.229 

0.328 

0.416 

0.494 

0.562 

0.620 

0.667 

0.703 

0.729 

0.745 

0.750 

7.00 

0.140 

0.267 

0.383 

0.486 

0.577 

0.656 

0.723 

0.778 

0.820 

0.851 

0.869 

0.875 

8.00 

0.160  0.306 

0.438 

0.555 

0.659 

0.750 

0.826 

0.889 

0.938 

0.972 

0.993 

.000 

9.00 

0.180  0.344 

0.492 

0.625 

0.742 

0.844 

0.930 

1.000 

1.055;  .094 

1.117 

.125 

10.00 

0.200  0.382 

0.547 

0.694 

O.S25 

0.937 

1.033 

1.111 

1.172;  .215 

.241 

.250 

11.00 

0.220  0.420 

0.602 

0.764 

0.907 

1.031 

1.136 

1.222 

1.289 

.337 

.365 

.375 

12  .00 

0.240;  0.458 

0.656 

0.833 

0.990 

1.125 

1.240 

1.333 

1.406 

.458 

.490 

.500 

13.00 

0.260  0.497 

0.711 

0.903 

1.072 

1.219 

1.343 

1.444 

1.523 

.580 

.641 

.625 

14.00 

0.280;  0.535 

0.766 

0.972 

1.154 

1.312  1.446 

1.556 

1.641 

.701 

.738 

1.750 

15.00 

0.300  0.573 

0.820 

1.042 

1.237 

1.406 

1.549 

1.667 

1.758 

1.823 

.862 

1.875 

16.00 

0.3201  0.611 

0.875 

1.111 

1.319 

1.500 

1.653 

1.778 

1.875 

1.944 

.986 

2.000 

17.00 

0.340  0.649 

0.930 

1.180 

1.402 

1.594 

1.756 

1.889 

1.992 

2.066 

2.110 

2.125 

18.00 

0.360  0.688 

0.984 

1.250 

1.484 

1.687 

1.859 

2.000 

2.109 

2.188 

2.234 

2.250 

19.00 

0.380  0.726 

1.039 

1.319 

1.567 

1.781 

1.963 

2.111 

2.227 

2.309 

2.358 

2.375 

20.00 

0.400!  0.764 

1.094 

1.389 

1.649 

1.875 

2.066 

2.222 

2.344 

2.431 

2.483 

2.500 

21  .00 

0.420  0.802 

1.148 

1.458 

1.732 

1.969 

2.169 

2.333 

2.461 

2.552 

2.607 

2.625 

22.00 

0.440;  0.840 

1.203 

1.528 

1.814 

2.062 

2.273 

2.444 

2.578 

2.674 

2.721 

2.750 

23.00 

0.460  0.879 

1.258 

1.597 

1.897 

2.156 

2.376 

2.556 

2.695 

2.795 

2.855 

2.875 

24.00 

0.480  0.917 

1.313 

1.667 

1.979 

2.250 

2.479 

2.667 

2.813 

2.917 

2.972 

3.000 

25.00 

0.500  0.955 

1.367 

1.736 

2.061 

2.344 

2.582 

2.778 

2.930 

3.038 

3.103 

3.125 

26.00 

0.520  0.993 

1.422 

1.806 

2.144 

2.437 

2.686 

2.889 

2.047 

3.160 

3.227 

3.250 

27.00 

0.540  1.031 

1.477 

1.875 

2.226 

2.531 

2.789 

3.000 

3.164 

3.281 

3.352 

3.375 

28  .00 

0.560'  1.070 

1.531 

1.944 

2.309 

2.625 

2.892 

3.111 

3.281 

3.403 

3.476 

3.500 

29.00 

0.580  1.108 

1.586 

2.014 

2.391 

2.719 

2.996 

3.222 

3.399 

3.524 

3.600 

3.625 

30.00 

0.600  1.146 

1.641 

2.083 

2.474 

2.812 

3.099 

3.333 

3.816  3.646 

3.724 

3.750 

The  upper  part  of  this  table  is  for  the  right  ascension,  and  the  lower  part  for  the 
declination ;  but  the  parts  are  interchangeable  by  moving  the  decimal  points. 

When  used  for  the  moon's  parallax,  P,  2h,  3ft  must  be  taken  out  in  the  table  ins 
the  columns  2ft,  4*,  6*,  etc. 


240 


TABLES 


Table  III.— FACTOR  FOB 
INTERPOLATION  GIVING 
COEFFICIENT  FOR  J0. 


Table  IV. — LOGARITHM    OF    THE 
EARTH'S  RADIUS. 


A! 
n 

n(n  —  1) 
1-2 

AI 
n 

18" 

0.00  — 

0.00000 

1.00 

.01 

0.00495 

0.99 

.02 

0.00980 

.98 

.03 

0.01455 

.97 

.04 

0.01920 

.96 

.05  — 

0.02375 

.95 

.06 

0.02820 

.94 

.07 

0.03255 

.93 

.08 

0.03680 

.92 

.09 

0.04095 

.91 

.10  — 

0.04500 

.90 

.11 

0.04895 

.89 

.12 

0.05280 

.88 

.13 

0.05655 

.87 

.14 

0.06020 

.86 

.15  — 

0.06375 

.85 

.16 

0.06720 

.84 

.17 

0.07055 

.83 

.18 

0.07380 

.82 

.19 

0.07695 

.81 

.20  - 

0.08000 

.80 

.21 

0.08295 

.79 

.22 

0.08580 

.78 

.23 

0.08855 

.77 

.24 

0.09120 

.76 

.25  — 

0.09375 

.75 

.26 

0.09621 

.74 

.27 

0.09855 

.73 

.28 

0.10080 

.72 

.29 

0.10295 

.71 

.30  — 

0.10500 

.70 

.31 

0.10695 

.69 

.32 

0.10880 

.68 

.33 

0.11055 

.67 

.34 

0.11220 

.66 

.35  — 

0.11375 

.65 

.36 

0.11520 

.64 

.37 

0.11655 

.63 

.38 

0.11780 

.62 

.39 

0.11895 

.61 

.40  — 

0.12000 

.60 

.41 

0.12095 

.59 

.42 

0.12180 

.58 

.43 

0.12255 

.57 

.44 

0.12320 

.56 

.45  — 

0.12375 

.55 

.46 

0.12420 

.54 

.47 

0.12455 

£3 

.48 

0.12480 

.52 

.49 

0.12495 

.51 

0.50  — 

0.12500 

0.50 

Lat. 

log  p. 

Lat. 

log  p. 

9.999 

9.999 

0° 

*000 

45° 

277 

r\/> 

1 

999 

46 

251 

26 

2 

998 

. 

47 

226 

Zo 

3 

996 

48 

200 

26 

4)  — 

4 

993 

49 

175 

zo 

5 

989 

50 

150 

25 

6 

7 
8 

984 
979 
972 

5 

7 

51 
52 
53 

125 

100 
076 

25 
25 
24 

9 

965 

. 

54 

051 

25 

10 

957 

55 

027 

24 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 

948 
938 
927 
916 
904 
891 
877 
862 
847 
831 
815 
798 
780 
761 
742 
723 
703 
682 
661 
639 
617 

10 
11 
11 

12 
13 
14 
15 
15 
16 
16 
17 
18 
19 
19 
19 
20 
21 
21 
22 
22 

99 

56 

57 
58 
59 
60 
61 
62 
63 
64 
65 
66 
67 
68 
69 
70 
71 
72 
73 
74 
75 
76 

004 
*980 
957 
934 
912 
890 
869 
848 
828 
808 
788 
770 
752 
734 
717 
701 
686 
671 
657 
644 
632 

23 
24 
23 
23 
22 
22 
21 
21 
20 
20 
20 
18 
18 
18 
17 
16 
15 
15 
14 
13 
12 

32 
33 
34 

595 
572 

548 

22 

23 
24 

77 
78 
79 

620 
609 
599 

11 
10 

35 

525 

94 

80 

590 

36 

501 

fnt 

81 

582 

37 

477 

24 

OC 

82 

574 

8 

38 

452 

25 

83 

568 

6 

39 

40 

428 
403 

25 

or 

84 

85 

562 
557 

5 

41 

378 

25 

oc 

86 

553 

4 

42 
43 
44 
45 

353 
327 
302 

277 

2o 
26 
25 
25 

26 

87 
88 
89 
90 

550 
548 
546 
546 

3 

2 
2 
0 

*  The  argument  of  Table  IV.  is  the 
geographical  latitude. 


TABLES. 


241 


Table  V. 


Table  VII. 


Table  VIII. 


!                              ^_ 

AMI.             log  MI'. 

_     Proportional 

6                                                               1—  —  ft1 

Parts.    MI- 

14°  W  40"  9.41780.8 

2.63828     .,,-,     0.0010000 

41             81.6 

lw.     0°  15'   0" 

397~~T^             0100 

42             82.4 

2  .     0     30     0 

2.62969    1^0             0200 

43             83.2 

3  .     0    45    0 

546    |J5             0300 

44            84.0 

4.1       00 

127     HI             0400 

5  .     1     15    0 

14    59    45    9.41784.8 

6  .     1     30    0 

2.61712    411             0500 

46             85.6 

7  .     1     45    0 

301     ;ii             0600 

47             86.4 

8.2      00 

2.60894     ;x{             0700 

48             87.2 

9  .     2    15    0 

490    T^n             0800 

49             88.0 

10  .    2    30    0 

090    ^g             0900 

2.59694    oQ9             1000 

14    59    50    9.41788.8 
51             89.6 

0  .1     0    1     30 

.2          30 

302    ^             1100 

52             90.4 

.3          4    30 

2.58913    oo^              1200 

53             91.2 

.4          60 

527     f£             1300 

54             92.0 

.5          7    30 

145     gj             1400 

.6          90 

14    59    55    9.41792.8 

.7        10    30 

2.57766    o7r             1500 

56             93.6 

.8        12      0 

391     o-o             1600 

57             94.4 

.9        13    30 

018    £S             1700 
2.56649    oftft             1800 

58             95.2 
59             96.0 

1  .0    0  15      0 

283    oSo             1900 

0  .01  0    V  9" 

15      0      0    9.41796.8 

.02        0  18 

2.55920             0.0012000 

1             97.6 

.03        0  27 

2QQ    A 

.04        0  36 

«7O.4 

3             99.3 

.05        0  45 

Table  VI. 

4    9.41800.1 

.06        0  54 

.07        1    3 

Mean  Sidereal  Equivalent 
Time.               in  Arc. 

15      0      5    9.41800.9 
6             01.7 
7             02.5 

.08        1  12 
.09        1  21 
0  .10  0    1  30 

8             03.3 

1*      15°     2'     37".85 

9             04.1 

0  .001     0  0".9 

2       30      4      55  .69 

.002        1  .8 

3       45      7      23  .54 
4       60      9      51  .39 

15      0    10    9.41804.9 
11             05.7 

.003        2  .7 
.004        3  .6 

5       75    12      19  .24 

12            06.5 

.005        4  .5 

6       90    14      47  .08 

13            07.3 

.006        5  .4 

7     105    17      14  .93 

14             08.1 

.007        6  .3 

8     120    19      42  .78 

.008        7  .2 

9     135     22      10  .62 
10     150    24      38  .47 

15      0    15    9.41808.9 
16             09.7 

.009        8  .1 
0  .010    0  9  .0 

11      165    27        6  .32 

17             10  6 

12     180    29      34  .17 
13     195    32        2  .01 

15      0     18    9.41811.4 

0  .0001  0  0".09 
.0002      0  .18 

14     210    34      29  .86 

.0003      0  .27 

15     225    36      57  .71 

AM.               log  AMI. 

.0004      0  .36 

16     240    39      25  .56 

14°  59'  40"     1.17593 

.0005      0  .45 

17      255    41      53  .40 

14    59    44       1.17595 

.0006      0  .54 

18      270    44      21  .25 

14    59    50       1.17600 

.0007      0  .63 

19     285    46      49  .10 

14    59    56       1.17605 

.0008      0  .72 

20     300    49      16  .95 

15      0      1       1.17610 

.0009      0  .81 

21     315    51      44  .80 

15      0      7       1.17615 

0.  0010  0  0.  90 

22     330    54      12  .64 

15      0    14       1.17620 

23     345    56      40  .49 

15      0    18       1.17623 

24     360    59        8  .33 

If) 


242 


TABLES. 


Table  IX. 


Table  X. 


Q. 

log  sin.  log  cos. 

0°  180°  360° 

-co   0.00000 

5  175  185  355 

8.94030  9.99834 

10  170  190  350 

9.23967  9.99335 

15  165  195  345 

9.41300  9.98494 

20  160  200  340 

9.53405  9.97299 

25  155  '205  335 

9.62595  9.95728 

30  150  210  330 

9.69897  9.93753 

35  145  215  325 

9.75859  9.91336 

40  140  220  320 

9.80807  9.88425 

45  135  225  315 

9.84949  9.84949 

50  130  230  310 

9.88425  9.80807 

55  125  235  305 

9.91336  9.75859 

60  120  240  300 

9.93753  9.69897 

65  115  245  295 

9.95728  9.62595 

70  110  250  290 

9.97299  9.53405 

75  105  255  285 

9.98494  9.41300 

80  100  260  280 

9.99335  9.23967 

85  95  265  275 

9.99S34  8.94030 

90    270 

0.00000    oo 

Q—  y. 

logi. 
7.6600          7.6675          7.6750 

0°     180°    360° 

15'.7ni   16'.0     .   16'.3     , 

5  175  185  355 

15.6     ,   15.9     .,  16.2    •* 

10  170  190  350 

15  .5    '«  15  .7    'o  16  .0    '« 

15  165  195  345 

15  .2    4  15  A    q  15  .7    'J 

20  160  200  340 

14.8    •;:  15.1     ft  15.3    '? 

25  155  205  335 

14  .2    a  14  .5    '2  14  .8    - 

30  150  210  330 

13.6    '?  13.8    •;  14.1    •' 

35  145  215  325 

12  .9    •'  13  .1    •'  13  .3    '° 

40  140  220  320 

12.0    •;  12.3,  ^  12  .5,'J 

45  135  225  315 

11  .1  TO  11  .3{^  11  .6  fa 

50  130  230  310 

10  .1  f  }[  10  .3  rr  10  .5  rx 

55  125  235  305 

Q    1              Q    9             Q    ^ 

HO  1.2    oJl.2    QI  1.2 

60  120  240  300 

7  .9  ,  9    8  .0    „    8  .1  1  q 

65  115  245  295 

6.7;'q          6.8        '2          6.8    {q 

r     A   1-6        tr      A      .4       'c      c   1-" 

70  110  250  290 

J'fl.3    J'7   .3    2*01-3 

75  105  255  285 

4>114    4>1     4    4'214 

80  100  260  280 

2.7-,'q         2.7        *q          2.8    / 

85     95  265  275 
90    90  270  270 

1  Art   1.4  1   i  if 

0  .0  *•     o  .0  •  :   o  .0  i> 

e  has  the  same  sign  as  cos  (Q  —  7). 


Table  XL 


Table  XII. 


h. 

log  sin  h. 

log  cos  h.  cosh. 

log  sec  h. 

5 

8.9403 

9.9983 

0.9924 

0.0017 

10 

9.2397 

9.9934 

0.9698 

0.0066 

15 

9.4130 

9.9849 

0.9330 

0.0151 

20 

9.5341 

9.9730 

0.8830 

0.0270 

25 

9.6259 

9.9573 

0.8214 

0.0427 

30 

9.6990 

9.9375 

0.7500 

0.0625 

35 

9.9586 

9.9134 

0.6710 

0.0866 

40 

9.8081 

9.8843 

0.5868 

0.1157 

45 

9.8495 

9.8495 

05000 

0.1505 

50 

9.8843 

9.8081 

0.4132 

0.1919 

55 

9.9134 

9.7586 

0.3290 

0.2414 

60 

9.9375 

9.6990 

0.2500 

0.3010 

65 

9.9573 

9.6259 

0.1786 

0.3741 

70 

9.9730 

9.5341 

0.1170 

0.4659 

75 

9.9849 

9.4130 

0.0670 

0.5870 

80 

9.9934 

9.2397 

0.0302 

0.7603 

85 

9.9983 

8.9403 

0.0076 

1.0597 

90 

0.0000 

CO 

0.0000 

CO 

All  the  quantities  in  Table  XL 
are  positive. 


0. 

togJI 

log«. 

Angle  of  the 
Vertical. 

0° 

0.00000   , 

0.00295   -. 

-0/      ^     9     0 

5 

01   J 

294    i 

2       °  ~1  55 

10 

04   J 

291    \ 

3r  K         -^  ^^ 
°           1   4Q 

15 

•    10  2 

285    2 

5    44      }  Jj 

20 

17    7 

278    Q 

7    23       {  f. 

25 

30 
35 

26  11 
37  11 

269  •; 

258] 
247  J  I 

8    48      {  ^ 
9    57      i  KT 
10    48      xXj 

40 

61    , 

234  \\ 

11    20      Qj| 

45 

50 

74  12 

221  }« 

209  ^ 

ii   2i+o^? 

55 

99  J3 

196  12 

10    50       JJJi 

60 
65 
70 

1U10 
121^ 

130   I 

184  10 
174™ 

165    Q 

9    59       V    Q 
8    50      JJ 

7    25 

75 

138    ° 

157    ° 

5    46       }  3|j 

80 

143    Q 

152    § 

3        57            ,     ry 

85 

146    f 

149    o 

2          0       i    (y      n 

90 

0.00147    ' 

0.00147    z 

—0      0  ^ 

TABLES. 
Table  XIIL,  GIVING  j/  IN  DEGREES  AND  DECIMAL. 

Top  Argument,  Natural  Sine  /3.    Side  Argument,  log  —  • 


243 


QD  cp  o»  o»  o»  o  o  M      a£  05  ^  id  *A 


«o  «o  «o  t-'  t>-  1-^  od  co  oi  os  as  o  o  T-H  I-H  c<i  <M'  co  co  co 


244 


TABLES. 


Table  XIV. — PART  L,  GIVING  Q  IN  DEGREES  AND  DECIMAL. 

Top  Argument,  v'  from  Table  XIII.    Side  Argument,  E. 


f- 

OrH^W 
rHrHrHrH 


e    00  C*  tO  O  •<  00  O9  CO 


rHrHrH<MC<J< 


OOO5i-H(NCOiOCOl>oo 
(MfNCCiCOCOCOCOCOCO"*1 


rH  CO  ^  1C  SO 


CO  rH  CO  CD  OS  rH 

5  CD  od  oi  o  c<5  co°  •HH  10  t>- 

D  CO  CO  CO  ^^  ^t^  ^f  ^^  '^  ^f 


Oi  (M  Tf  l>  O5  rH 


(M(M(MCOCOCOCOCCCO 


t-QO 
COCC 


rH  C^  CO  ^  CD" 


•^'t^CXJOi-J 
(MIMlMCOCd 


lMCOCdCOCOCOCOCO 


cO 
CO 


O  rH 
Tfl  TJH 


c^c^ 


C<j  Tj<  CD  OS  rH  CO 
GO  O^  O  C^  CO  *f  *O  1^*  OO  O5  O  rH  CO  ^* 
C^  C^  CO  CO  CO  CO  CO  CO  CO  CO  ^*  ^f  ^  ^f 


lOCOt^QOOrHC<i 
(MfNlMC^COCCCO 


>^  00  O5  rH  C<j  CC 

CO  CO  CO  "t1  ^  Tj< 


i— ICOiC:t^QOO!MT^iO 


tO  l>-  CO  Oi  O  rH 

COCO  COCO  rti  ^ 


rH  <M  "f  iO  t^  QO  Oi 


^  5^ 


rHrHrHrHrHrHrHrHC<l<M 


«O  t>-  CO  O  i—  I 

CO  CO  CO  CO  CO 


Oi-HCOT*«cOt-aOO»—  I 


O5  O  rH  <N  CO  >O 


CO  CO  CO  CO  CO  CO 


•      •      •      •      •      •      • ^-      •      * 

rHrHrHrHrHrHrHrHC^S^l 


CO  -^  lO  CD  t>-  CO 
CO  CO  CO  CO  CO  CO 


Oi—  ! 


!>•  GO  O5  OS  O  rH 

co  rf  io  t^  oo 

CO  CO  CO  CO  CO  CO 


Or-  i 


0  rH  0\  •*#  ItS  CC5  tr^  00  OS  O 


rH  <M  C-l  CO  ^  TJJ 

C<i  CO  -rJH  »O  O  t>^ 
CO  CO  CO  CO  CO  CO 


O  •—  IrH 


CD  CD  CD  I>-  t>  GO 
^OOOiO    rHC<icO'HH'OCD' 

•    -.  I  co  co  co  co  co  co 


O  rH  rH  rH  <M  C<l 

CO  CO  CO  CO  CO  CO 


iC  kO  iO  «D  CD  CD 
O  CO  CO  CO  CO  CO 


O   rH   C<i   CO'   Tf  lO 

CO  CO  CO  CO  CO  CO 


O  rH  C<I  CO  Tf«  lO 

CO  CO  CO  CO  CO  CO 


TABLES. 


245 


Table  JTJF.— PART  II.,  GIVING  Q  IN  DEGREES  AND  DECIMAL. 

Top  argument,  v'  from  Table  XIII.    Side  argument,  E. 


^  (N  0  00  »0 


rH  <-<  i—  i  CM  <M  (M 


(M  O5  1C  O5  CO  «O 
r-i  -^  «D  OS  r-i 
CO  CO  CO  W* 


•^  O5  Tfl  GO  rH  CO  Tft 


£i  §1  co  co  co  co  55 


?c4io'oo*ocoiooooco 


»OOOO<MiOt^O5i—  i 
<M(MCOCOCOCOCO^f 


o  no  o  «    o  TJH  os  co  «    o 


<MC^<MCOCOcbcOCO'«tl 


coiCi>o<-HCcicta5i- 

(MIMlMCNCOCOCOCOCOTti 


J<1  "^  CO  OO  ^^  C^  TjH  CO  t^»  Oi 

(M(M(M(NCOCOCOCOCOCO 


I  CO  CO  CO  CO  CO 


OOrHi—  l 

o  <?i  -^  «c3  od  o  <M'  ^'  «o  od 


^JlC^JNC^COCOCOCO 


10  C^O 

^*^' 


^ 


rH  00  10  rH 

fei?: 


CO  1C  rH  t>-  r-i 


t>.  G<J  CO  CO  Oi  10  O 
«C>  t>^  Oi 
CO  CO  CO  - 


rHJ^COQO-^OlOrHCDrH 


cocococococoTr^^'^ 


-H  TJJ  00  O 

^  ^^§ 


O  rH  CO*  ^  5C5  t^-  O: 
<M  <M  (M  <N  C^  (M  (M 


CO  CO  CO  CO  CO  CO  CO 


OS  (N  iO  Oi  CO  O 

•«^  co  t>^  od  o  i-i 

rt<  rj<  rt<  •*  »0  »0 


CO  rH  1C  00  rH  Tf 


C^  C^  (?1  C<l  <M  C^ 


8  CO 


COCOCOCOCOCOCO'^'^f 


OrHCCrfllOl>COOi—  (CO 


O  iM  CO  TJH  CO  t^ 
(M  (M  <N  (M  <M  <M 


M  CO  CO  CO 


t^  rH  T**  t^  O  <M 


CO  CO  CO  CO  "^  rfi 


% 


rt<  iO  CD  00  O> 


CO  CO  COCO  CO  CO 


246 


TABLES. 


Table  XV. — WHOLE  NUMBER  OF  ECLIPSES  IN  ONE  SAROS, 

1889-1906. 


Vpftr 

Total  Species. 

Annular  Species. 

Lunar  Eclipses. 

In 

1  LHi  . 

Total. 

Partial. 

Annular. 

Partial. 

Total. 

Partial. 

Year. 

1889 

2 

1 

2 

5 

1890 

— 

2 

1 

3 

1891 



1 

1 

2 

4 

1892 

1 

1 

1 

1 

4 

1893 

1 

1 



2 

1894 

1 

1 

2 

4 

1895 

2 

1 

2 

5 

1896 

1 

1 

2 

4 

1897 





2 



2 

1898 

1 

1 

1 

1 

2 

6 

1899 

1 

1 

1 

1 

1 

5 

1900 

1 

1 

1 

3 

1901 

1 

1 

1 

3 

1902 

2 

1 

2 

5 

1903 

1 

1 

2 

4 

1904 

1 

1 



2 

1905 

1 

1 

2 

4 

1906 

— 

3 

2 

5 

Species  in 
one  Saros. 

12 

6 

16 

8 

11 

17 

70 

NOTE. — The  dash  denotes  no  eclipse  of  the  species  during  the  year.  Lunar  Appulse. 

1890,  and  1901. 


THE  APPROXIMATE  LOCATIONS  OF  FUTURE  ECLIPSES. 

This  may  be  done  with  little  or  no  trouble  as  follows : 

Take  the  eclipse  1904,  Sept.  9,  tf  8A  49™.  Applying  the  Saros  on  the  next  page 
in  which  we  must  take  11  days,  we  have  for  the  date  and  approximate  time  of  con- 
junction 1922,  Sept.  20,  16*  31" 

The  eclipse  At  Noon,  Art.  114,  occurred  in  longitude  -f  133  5  W.,  and  latitude 
—  4°  35'  south.  During  the  7*  42m  of  the  Saros  the  earth  will  revolve  through  115.6 
degrees  (see  next  page).  The  Eclipse  At  Noon  will  be  in  longitude  +249  W.,  or 
111  East.  By  the  table  on  the  next  page,  column  6,  the  series  is  seen  to  be  moving 
South.  This  motion  averages  about  5°  at  each  appearance,  so  that  the  eclipse  At 
Noon  will  be  in  latitude  —  9°  35'  South.  In  Art,  45  we  ascertained  that  the  incli- 
nation of  the  path  is  increasing  slightly,  which  will  cause  the  beginning  to  fall  only 
2°  or  3°  south,  while  the  ending  will  be  7°  or  8°  south  of  1904. 

By  the  present  table,  column  5,  we  see  that  the  direction  of  the  path  is  south,  and 
in  column  3  that  the  duration  in  column  2  is  decreasing,  so  that  it  will  be  somewhat 
less  than  6™  24*  at  its  greatest  appearance.  These  results  are  only  approximate,  for 
mean  values  alone  can  be  used. 


TABLES. 


247 


Table  XVI. — LIST  OF  ALL  TOTAL  ECLIPSES  IN  ONE  SAROS, 
1889  to  1906. 


Date. 

Duration. 

Increasing 
or 
Decreasing. 

At  Noon. 

<(>            0) 

Direction 
of  Path. 

Motion 
of 
Series. 

TO         S 

o        o 

1889,  Jan.  1 
1889,  Dec.  21 

2     15 
4     15 

D 
D 

+  37  138  W. 
—  14    73 

#  slightly. 

N 

N 

1892,  April  26 

4     17 

D 

—  64  139 

N 

S 

1893,  April!  6 

4    46 

D 

—   1     37 

N 

S 

1894,  Sept.  28 

0     11 

I 

—  34  274 

S 

N 

1896,  Aug.  8 

2    42 

D 

+  65  248 

S 

N 

1898,  Jan.  21 

2     19 

I 

+  13  291 

N 

S 

1900,  May  28 

2      9 

D 

+  45    45 

N 

N 

1901,  May  17 

6     27 

I 

—   2  263 

N 

N 

1903,  Sept.  20 

2     15 

D 

—  70  259 

S 

S 

1904,  Sept.  9 

6     24 

D 

—   5  133 

S 

S 

1905,  Aug.  29 

3    45 

1 

+  46    12 

S 

S 

1906.     All  the  Solar  Eclipses  will  be  Partial. 

LIST  OF  ALL  ANNULAR  ECLIPSES  IN  ONE  SAROS,  1889  TO  1906. 


Date. 

Duration. 

Increasing 
or 
Decreasing. 

At  Noon. 

*            0, 

Direction 
of  Path. 

Motion 
of 
Series. 

m      s 

0             0 

1889,  June  27 

7     23 

D 

—  10   313 

N 

S 

1890,  June  16 

4     12 

J 

—  37    330 

N 

s 

1890,  Dec.  11 

0    17 

I 

—  54   230 

S 

N 

1891,  June  6 

Not  given. 

I 

+  70   250 

N 

S 

1893,  Oct.  9 

3     46 

D 

-j-13   126 

S 

N 

1894,  April  5 

0       1 

1 

+  47    246 

N 

S 

1896,  Feb.  13 

Not  given. 

D 

S.  None. 

N 

S 

1897,  Feb.  1 

2    38 

D 

—  29   118 

N 

s 

1897,  July  29 

1       6 

I 

+  15     58 

S 

N 

1898,  July  18 

6    13 

1 

—  43   120 

S 

N 

1899,  Dec.  2 

Not  given. 

D 

—  88    198 

S 

S 

1900,  Nov.  21 

0    44 

D 

—  33   294 

s 

S 

1901,  Nov.  10 

11       3 

I 

+  12    294 

s 

s 

1903,  Mar.  28 

1    49 

D 

+  65   210 

N 

N 

1904,  Mar.  16 

8      3 

D 

+    6    264 

N 

N 

1905,  Mar.  5 

7    58 

I 

—  43   250 

N 

N 

The  Saros  6585.321 222  days,  or 

18"  ]^d  7*  42™  33.6s,  if  Feb.  29  intervenes 
Kevolution  of  the  earth  in  7ft  42™  33.68  =  115°  6'  west. 


times. 


BY    THE    SAME}   AUTHOR. 


TREATISE 


PROJECTION  OFTH_E  SPHERE 


WITH   PRECEPTS  AND   TABLES 


FOR    FACILITATING  THE 


ORTHOGRAPHIC  AND  STEREOGRAPH  1C 
PROJECTIONS 

OF  ANY  PORTION  OF  THE  EARTH'S  SURFACE 

BY  ROBERDEAU    BUCHANAN,  S.  B. 

Assistant  in  the  Office  of  the  American  Ephemeris  and  Nautical  Almanac 

AT 
WASHINGTON,  D.  C.,  1890 

With  plates,  pp.  47 

In  this  work  the  method  given,  with  formulae  and  tables,  was  devised  by  the  author  for  his  own 
use,  and  has  been  used  by  him  yearly  in  projecting  solar  eclipses  and  in  connection  with  other  duties  in 
the  office  of  the  American  Ephemeris  and  Nautical  Almanac. 

PRIVATELY  PRINTED.     PRICE,  75  CENTS 

MAY  BE  HAD  BY  APPLYING  TO  THE  AUTHOR,  201$  Q  STREET,  WASHINGTON,  D.  C. 


,10 


Plate! . 
Fig.2. 


Total  Eclipse 
1904,  September  9. 


Plate  II 
Fig.  3. 


H 


Total  Eclipse 
1860,  July  18. 


Diagram  sTwwing  the  relations 

between  the 

Fundamental  Plane  and  tfrnoralCone 
in  their  extreme  positions. 


Moon's  Parallax 
and  quantities 
depending  upon  it. 


Greatest  Parallax.      61' 28*8 


Umbral  C 


(two  post 
and  its  extn 
<  —.016 


Least  Parallax.       S3'  55."<f  +.029 

*  =  63.76 

z  =  Distance  of  Moon  from  Fundamer* 

sin  f.=  Distance  of  vertex  of  Cone  from  t/ 

sin  f=  Distance  of  vertex  of  Cone  above 

Vertical  Scale :  Unit =0.5  inch.    Sorizonta 


Plate  III  Fig.4 


The    UmbralCone. 


Sun's  distance 
and  quantities 

s)  depending  upon  it. 

radii. 


_.Least    istance 


\ApU. 


OctJ.-\ 


v 


Greatest  distance 


Mean  Fundamental  Plane. 


B 


Note:- The  anole  of  the  Cone 
is  Tnuch  exaggerated  in 
this  figure. 


\Plane. 
foon. 

kidamental  Plane. 
:  0.01  =  0.5 inch. 


J^B.del. 


Peculiarities  of  Eclipses 

Plate  I Y  Fig.  7 

H  The  point  li  of  the  southern 
limiting  curve  has  a  greater 
latitude  than,  the  correspond- 
ing northern  point. 


Small  central^ 


E  Total -Annular  Eclipse. 


E  Annular  at  the  ends 


DjVext  eclipse  in 
the  series  to^. 
£u£  one  branch 
to  rising  a> 
setting  curve. 

B  Partial  eclipse,  small 
branch  to  rising  and 
setting  curve. 

B 

C  Central 
eclipse  of  the 
midnight  sun. 


A.The  rising  and  setting  Q  77^,  nearest  point  of  the 

curve  has  nearly  separated  into         eclipse  to  the  South  Pole,  is 
two  branches  at  a.     _ff,S,  del.        the Jfgrthern limiting  curve. 


e 


Plate  V 

Fig.  8 


,,--'    Total  Eclipse 
1904,  September  9. 


TOTAL  ECLIPSE 


OF 


Longitude 


East  of 


Greemvlch 


Note:77w?  hours  of  beginning  and  endi, 


P TE  MBE R  9T^  19  O  4  Plate  VI,  Fig.  IT. 


<§>  longitude    &          West.        of 


Greemvnch 


:  expressed  in  Grcenwicli.Ne.an  Time. 


. 


• 


TOTAL  ECLIPSE 


longitude         Ea*t 


AUGUST  8™  1896.         Plate  VII,  Fig.  18. 


if         Greenwich 


f\ 


\ 


Plate  VIII 
Fig.  21. 


Total  Eclipse 
1904,  September  9. 


13 


Total  Eclipse 

19O4,  September  9. 

Shape  of  the  Shadow  upon  the  Earth. 


Plate  IX 
Fig.  23 


Plate  XI 


M 


J&S.del. 


Limiting  Parallels. 


YC  22103 


